forked from 1b-agent/fieldcraft-upstream
Planar all-quad meshing: size fields, TFI mapped meshes, free quad mesher (M5 base) (#18)
* Planar all-quad meshing: size fields, TFI mapped meshes, free quad mesher, planar quad validity - modelcraft_mesh::size: SizeField trait, division counts, graded polyline division, per-loop parity - modelcraft_mesh::quad: transfinite (Gordon-Hall) mapped meshes of four-sided regions; free all-quad meshes of polygons with holes on prescribed boundary nodes (half-resolution CDT with Ruppert refinement, Lee-Lo recombination, uniform split, smart Laplacian plus Freitag-Plassmann untangling), exact validity gate on every element - modelcraft_mesh::quality: Robinson aspect ratio and skew for quads, exact planar quad validity Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * quad mesher: true-boundary guard for Steiner points, untangling by corner area, parity retry, recombination quality floor; I-beam and perforated plate tests Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * docs: quad meshing papers (B7), roadmap M5 and M2, architecture pipeline step Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm * quad mesher: simplify (shared corner and validity helpers, vertex rotation helper, guard decoupled from the CDT, verify inside the CDT, non-allocating final gate) Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm --------- Co-authored-by: jyviko <[email protected]> Co-authored-by: Claude Opus 5.5 <[email protected]>
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@@ -20,10 +20,10 @@ hex meshes today and it gives the best elements where stresses are read.
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| M0 Foundation | Workspace, quality gates, FE model, command engine with rollback and panic guard, CLI, geometry boundary on SolidCraft | done |
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| M1 Geometry in and surface preparation | Open `.solidcraft` designs with persistent face names, assemblies placed (done); face-tagged 3MF (done); test corpus, `cargo xtask corpus` (done); exact predicates (done); quality metrics and certified validity gate (done); weld, closed/manifold/orientation and self-intersection checks, feature edges (`modelcraft-mesh::surface`, `mesh.surface.check`, done) | in progress |
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| M2 Tet meshing and size field | Size field (global, per face and edge seeds, curvature and proximity); Bowyer–Watson Delaunay (exact, symbolic perturbation, verified; `modelcraft-mesh::delaunay`, done); robust CDT boundary recovery (Diazzi et al. 2023); Delaunay refinement, ODT smoothing, sliver removal; tet4 and tet10 with mid-side nodes on the B-rep; mesh verify per set | in progress |
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| M2 Tet meshing and size field | Size field (global, per face and edge seeds, curvature and proximity; the `modelcraft_mesh::size::SizeField` trait and edge-division helpers are in); Bowyer–Watson Delaunay (exact, symbolic perturbation, verified; `modelcraft-mesh::delaunay`, done); robust CDT boundary recovery (Diazzi et al. 2023); Delaunay refinement, ODT smoothing, sliver removal; tet4 and tet10 with mid-side nodes on the B-rep; mesh verify per set | in progress |
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| M3 First deck | Sets and surfaces bound to SolidCraft face, edge, vertex and body names, resolved after meshing; materials (isotropic and orthotropic elastic, density) with a unit-system label and consistency check; sections (solid, shell, beam profiles) assigned by set; element formulation per region mapped to each solver; BCs (fixed, displacement, symmetry); loads (force and moment on reference points, pressure, gravity); reference points with kinematic and distributing coupling; steps (linear static, frequency) with explicit per-step load and BC tables and output presets; writers for Abaqus input, Nastran SOL 101/103 and LS-DYNA implicit; the `.modelcraft` file | planned |
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| M4 Own solver and first results | Linear static and modal solver (tet4/10, hex8, hex20; sparse direct, Lanczos), verified against analytic solutions; deformed shape, contours, mode shapes, probes, reaction sums | planned |
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| M5 Structured hex | 2D quad meshing of faces; mapped hex of six-sided regions; sweep meshing; conforming interfaces and tet–hex transition; automatic decomposition into sweepable and mappable regions, proposed as SolidCraft splits | planned |
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| M5 Structured hex | 2D quad meshing of faces (`modelcraft-mesh::quad`: transfinite mapped meshes, free all-quad meshes of polygons with holes on prescribed boundary nodes, exact planar quad validity, shared `SizeField` and even edge divisions; done for planar faces); mapped hex of six-sided regions; sweep meshing; conforming interfaces and tet–hex transition; automatic decomposition into sweepable and mappable regions, proposed as SolidCraft splits | started |
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| M6 Automatic hex | Octree/grid-dual all-hex for any part; hex-dominant by tet recombination; polycube for blocky parts; smoothing, untangling, boundary padding, layers through thin walls; mesh editing and orphan-mesh import | planned |
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| M7 Assemblies and contact | Part instances from SolidCraft occurrences; tie between non-matching meshes; general and pairwise contact with automatic pair detection, friction; bolt pretension, springs, point masses; amplitudes, multiple steps; nonlinear static steps in decks | planned |
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| M8 Materials, explicit, shells, composites | Plasticity, hyperelasticity, damage data per solver; explicit dynamics decks (Abaqus/Explicit, LS-DYNA); midsurface shells; composite layups and material orientations, continuum shells on swept hex; heat transfer | planned |
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@@ -8,7 +8,9 @@
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#![forbid(unsafe_code)]
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pub mod delaunay;
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pub mod quad;
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pub mod quality;
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pub mod size;
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pub mod surface;
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mod vec3;
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@@ -117,6 +117,13 @@ pub fn is_admissible(x: f64) -> bool {
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x == 0.0 || (x.abs() >= MIN_MAGNITUDE && x.abs() <= MAX_MAGNITUDE)
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}
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/// `x` with a magnitude below [`MIN_MAGNITUDE`] rounded to zero (tessellation and construction
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/// noise); other values unchanged. The result is admissible when `|x| <= MAX_MAGNITUDE`.
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#[inline]
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pub fn snap_to_domain(x: f64) -> f64 {
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if x.abs() < MIN_MAGNITUDE { 0.0 } else { x }
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}
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fn check<const N: usize>(c: [f64; N]) -> Result<[f64; N]> {
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match c.iter().find(|x| !is_admissible(**x)) {
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None => Ok(c),
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@@ -0,0 +1,775 @@
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//! Planar constrained Delaunay triangulation with Delaunay refinement to a size field.
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//!
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//! - Insertion: Bowyer–Watson (Bowyer, Comput. J. 24(2), 1981; Watson, Comput. J. 24(2), 1981)
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//! inside a super-triangle, with exact `orient2d`/`incircle` (Shewchuk 1997). The cavity grows
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//! from the triangle containing the point and never crosses a constrained edge, which is the
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//! constrained Delaunay insertion of Shewchuk, *Delaunay refinement algorithms for triangular
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//! mesh generation*, CGTA 22:21–74, 2002, §2.
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//! - Constraint recovery: Sloan, *A fast algorithm for generating constrained Delaunay
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//! triangulations*, Computers & Structures 47(3):441–450, 1993: flip the edges crossing a
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//! segment until it appears, then restore the Delaunay property with Lawson flips.
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//! - Refinement: Ruppert, *A Delaunay refinement algorithm for quality 2-dimensional mesh
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//! generation*, J. Algorithms 18(3):548–585, 1995 (circumcentres of triangles that are too
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//! large for the size field or have a circumradius-to-shortest-edge ratio above √2), with one
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//! change: boundary segments carry prescribed nodes and are never split, so a circumcentre that
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//! falls outside the region, comes closer than a fraction of the local size to a vertex, or is
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//! refused by the caller's guard (clearance from the true boundary) is not inserted (a
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//! too-large triangle gets its centroid instead). Ruppert's encroachment rule (diametral
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//! circles) would forbid any refinement next to a boundary coarser than the size field; the
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//! clearances keep the packing argument for termination.
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//!
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//! Every loop is bounded (walk steps, cavity rounds, flips, refinement passes and the triangle
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//! count), and a cavity is checked before it is retriangulated: the new triangles all have
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//! positive exact orientation and no constrained edge is removed.
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use std::collections::VecDeque;
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use super::domain::Region;
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use crate::predicates::{Point2, Sign, incircle, orient2d, snap_to_domain};
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use crate::size::{SizeField, checked_size};
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use crate::{Error, Result};
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const NONE: u32 = u32::MAX;
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/// Rounds of cavity shrinking before an insertion is rejected.
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const MAX_CAVITY_ROUNDS: usize = 32;
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/// Refinement passes over the triangles.
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const MAX_REFINE_PASSES: usize = 64;
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#[derive(Debug, Clone, Copy)]
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struct Tri {
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/// Counterclockwise vertices.
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v: [u32; 3],
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/// `n[i]`: the neighbour across the edge opposite `v[i]` (from `v[i+1]` to `v[i+2]`).
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n: [u32; 3],
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/// Whether the edge opposite `v[i]` is a constrained (boundary) edge.
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fixed: [bool; 3],
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inside: bool,
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alive: bool,
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}
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/// The triangles of the region: counterclockwise vertex triples into `pts`.
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#[derive(Debug, Clone)]
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pub(crate) struct Triangulation {
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pub pts: Vec<[f64; 2]>,
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pub p2: Vec<Point2>,
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pub tris: Vec<[u32; 3]>,
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}
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pub(crate) struct Cdt {
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pts: Vec<[f64; 2]>,
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p2: Vec<Point2>,
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tris: Vec<Tri>,
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free: Vec<u32>,
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vt: Vec<u32>,
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last: u32,
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mark: Vec<u32>,
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epoch: u32,
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max_tris: usize,
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}
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/// Steiner points keep this fraction of the local size from vertices (and, through the
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/// caller's guard, from the boundary).
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pub(crate) const CLEARANCE: f64 = 0.3;
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/// A constructed point snapped into the predicates' domain, or `None` when it is outside the
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/// model's coordinate range.
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fn steiner_point(c: [f64; 2]) -> Option<Point2> {
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let c = [snap_to_domain(c[0]), snap_to_domain(c[1])];
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if !(c[0].abs() <= modelcraft_model::MAX_COORD && c[1].abs() <= modelcraft_model::MAX_COORD) {
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return None;
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}
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Point2::new(c[0], c[1]).ok()
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}
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fn err(what: &str) -> Error {
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Error::Failed(format!("triangulation: {what}"))
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}
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/// The edge index `i` of `tri` that runs from `a` to `b` (opposite `v[i]`).
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fn edge_index(tri: &Tri, a: u32, b: u32) -> Option<usize> {
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(0..3).find(|&i| tri.v[(i + 1) % 3] == a && tri.v[(i + 2) % 3] == b)
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}
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/// Every edge of `tri` keeps `p` on its left or on it (exact).
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fn contains(tri: &Tri, p2: &[Point2], p: &Point2) -> bool {
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(0..3).all(|i| match (p2.get(tri.v[(i + 1) % 3] as usize), p2.get(tri.v[(i + 2) % 3] as usize)) {
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(Some(a), Some(b)) => orient2d(a, b, p) != Sign::Negative,
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_ => false,
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})
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}
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impl Cdt {
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fn t(&self, t: u32) -> Result<&Tri> {
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self.tris.get(t as usize).ok_or_else(|| err("triangle index out of range"))
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}
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fn tm(&mut self, t: u32) -> Result<&mut Tri> {
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self.tris.get_mut(t as usize).ok_or_else(|| err("triangle index out of range"))
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}
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fn p(&self, v: u32) -> Result<&Point2> {
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self.p2.get(v as usize).ok_or_else(|| err("vertex index out of range"))
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}
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fn xy(&self, v: u32) -> Result<[f64; 2]> {
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self.pts.get(v as usize).copied().ok_or_else(|| err("vertex index out of range"))
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}
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/// Triangulate `region`'s boundary points (ids preserved) and recover its loops as
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/// constrained edges; triangles inside the region are marked.
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pub fn build(region: &Region, max_tris: usize) -> Result<Cdt> {
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let n = region.pts.len();
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let (lo, hi) = super::domain::bbox(region.pts.iter().copied());
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let c = [0.5 * (lo[0] + hi[0]), 0.5 * (lo[1] + hi[1])];
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let d = (hi[0] - lo[0]).max(hi[1] - lo[1]).max(1.0);
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let corners = [[c[0] - 20.0 * d, c[1] - 10.0 * d], [c[0] + 20.0 * d, c[1] - 10.0 * d], [c[0], c[1] + 20.0 * d]];
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let mut pts = region.pts.clone();
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let mut p2 = region.p2.clone();
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for k in corners {
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let k = [snap_to_domain(k[0]), snap_to_domain(k[1])];
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p2.push(Point2::new(k[0], k[1])?);
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pts.push(k);
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}
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let s = n as u32;
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let mut cdt = Cdt {
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vt: vec![NONE; pts.len()],
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pts,
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p2,
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tris: vec![Tri { v: [s, s + 1, s + 2], n: [NONE; 3], fixed: [false; 3], inside: false, alive: true }],
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free: Vec::new(),
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last: 0,
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mark: vec![0],
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epoch: 0,
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max_tris,
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};
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for v in [s, s + 1, s + 2] {
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if let Some(x) = cdt.vt.get_mut(v as usize) {
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*x = 0;
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}
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}
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// Insert in Morton order (short walks), ties by id.
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let span = [(hi[0] - lo[0]).max(f64::MIN_POSITIVE), (hi[1] - lo[1]).max(f64::MIN_POSITIVE)];
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let key = |p: &[f64; 2]| {
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let q = |x: f64, l: f64, s: f64| ((((x - l) / s) * 65535.0) as u32).min(65535);
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let (mut x, mut y) = (q(p[0], lo[0], span[0]), q(p[1], lo[1], span[1]));
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let mut k = 0u64;
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for bit in 0..16 {
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k |= u64::from(x & 1) << (2 * bit) | u64::from(y & 1) << (2 * bit + 1);
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x >>= 1;
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y >>= 1;
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}
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k
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};
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let mut order: Vec<(u64, u32)> = region.pts.iter().enumerate().map(|(i, p)| (key(p), i as u32)).collect();
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order.sort_unstable();
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for (_, v) in order {
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if !cdt.insert(v, None)? {
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return Err(err("a boundary point could not be inserted"));
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}
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}
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for l in ®ion.loops {
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for (i, &a) in l.iter().enumerate() {
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let b = l.get((i + 1) % l.len()).copied().ok_or_else(|| err("loop"))?;
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cdt.recover(a, b)?;
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}
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}
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cdt.classify()?;
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#[cfg(test)]
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assert_eq!(cdt.verify(), Ok(()), "after build");
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Ok(cdt)
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}
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fn alloc(&mut self, tri: Tri) -> Result<u32> {
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if let Some(t) = self.free.pop() {
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*self.tm(t)? = tri;
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return Ok(t);
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}
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if self.tris.len() >= self.max_tris {
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return Err(Error::Limit(format!("the triangulation exceeds {} triangles", self.max_tris)));
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}
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self.tris.push(tri);
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self.mark.push(0);
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Ok(self.tris.len() as u32 - 1)
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}
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/// The triangle containing `p` (on its boundary counts), by a visibility walk from `start`
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/// with a linear scan as the fallback.
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fn locate(&self, p: &Point2, start: u32) -> Result<u32> {
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let mut t = if self.t(start).is_ok_and(|x| x.alive) { start } else { self.any_alive()? };
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let cap = 64 + 4 * self.tris.len();
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'walk: for step in 0..cap {
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let tri = *self.t(t)?;
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for k in 0..3 {
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let i = (k + step) % 3;
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let (a, b) = (tri.v[(i + 1) % 3], tri.v[(i + 2) % 3]);
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if orient2d(self.p(a)?, self.p(b)?, p) == Sign::Negative {
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let next = tri.n[i];
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if next == NONE {
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return Err(err("point outside the super-triangle"));
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}
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t = next;
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continue 'walk;
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}
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}
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return Ok(t);
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}
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self.tris.iter().position(|tri| tri.alive && contains(tri, &self.p2, p)).map(|i| i as u32).ok_or_else(|| err("point location failed"))
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}
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fn any_alive(&self) -> Result<u32> {
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self.tris.iter().position(|t| t.alive).map(|i| i as u32).ok_or_else(|| err("no triangles"))
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}
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/// Insert vertex `v` (already in `pts`). Boundary points (`steiner == None`) must be inserted.
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/// Steiner points (`Some(min_distance)`) may be rejected (outside the region, closer than
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/// `min_distance` to a vertex, or no valid cavity), leaving the triangulation unchanged;
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/// returns whether the point went in.
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fn insert(&mut self, v: u32, steiner: Option<f64>) -> Result<bool> {
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let p = *self.p(v)?;
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let t0 = self.locate(&p, self.last)?;
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let tri0 = *self.t(t0)?;
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for &u in &tri0.v {
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if self.p(u)?.coords() == p.coords() {
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return match steiner {
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Some(_) => Ok(false),
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None => Err(err("repeated point")),
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};
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}
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}
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if steiner.is_some() && !tri0.inside {
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return Ok(false);
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}
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let mut excluded: Vec<u32> = Vec::new();
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for _ in 0..MAX_CAVITY_ROUNDS {
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let (cavity, edges) = self.cavity(t0, &p, &excluded)?;
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let mut bad: Vec<u32> = Vec::new();
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// A constrained edge inside the cavity: drop the triangle on the far side from p.
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for &t in &cavity {
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let tri = *self.t(t)?;
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for i in 0..3 {
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if tri.fixed[i] && tri.n[i] != NONE && self.marked(tri.n[i]) {
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let (a, b) = (tri.v[(i + 1) % 3], tri.v[(i + 2) % 3]);
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if orient2d(self.p(a)?, self.p(b)?, &p) != Sign::Positive {
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bad.push(t);
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}
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}
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}
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}
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// Every new triangle (a, b, p) must be counterclockwise.
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for &(a, b, _, _, owner) in &edges {
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if orient2d(self.p(a)?, self.p(b)?, &p) != Sign::Positive {
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bad.push(owner);
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}
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}
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if let Some(min_distance) = steiner {
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let pc = p.coords();
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for &(a, _, _, _, _) in &edges {
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let pa = self.xy(a)?;
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if (pa[0] - pc[0]).powi(2) + (pa[1] - pc[1]).powi(2) < min_distance * min_distance {
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return Ok(false);
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}
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}
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}
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if bad.is_empty() && edges.len() == cavity.len() + 2 {
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self.retriangulate(v, &cavity, &edges, tri0.inside)?;
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return Ok(true);
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}
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if bad.contains(&t0) || bad.is_empty() {
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break;
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}
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excluded.extend(bad);
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excluded.sort_unstable();
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excluded.dedup();
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}
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match steiner {
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Some(_) => Ok(false),
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None => Err(err("no valid cavity for a boundary point")),
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}
|
||||
}
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fn marked(&self, t: u32) -> bool {
|
||||
self.mark.get(t as usize).is_some_and(|&m| m == self.epoch)
|
||||
}
|
||||
|
||||
/// The Bowyer–Watson cavity of `p` from `t0` (never crossing constrained edges, skipping
|
||||
/// `excluded`), marked with the current epoch, and its boundary edges `(a, b, outside
|
||||
/// neighbour, constrained, owner)` with the cavity on their left.
|
||||
#[allow(clippy::type_complexity)]
|
||||
fn cavity(&mut self, t0: u32, p: &Point2, excluded: &[u32]) -> Result<(Vec<u32>, Vec<(u32, u32, u32, bool, u32)>)> {
|
||||
self.epoch = self.epoch.wrapping_add(1);
|
||||
if self.epoch == 0 {
|
||||
self.mark.iter_mut().for_each(|m| *m = 0);
|
||||
self.epoch = 1;
|
||||
}
|
||||
let epoch = self.epoch;
|
||||
let set = |mark: &mut Vec<u32>, t: u32| {
|
||||
if let Some(m) = mark.get_mut(t as usize) {
|
||||
*m = epoch;
|
||||
}
|
||||
};
|
||||
set(&mut self.mark, t0);
|
||||
let mut cavity = vec![t0];
|
||||
let mut k = 0;
|
||||
while let Some(&t) = cavity.get(k) {
|
||||
k += 1;
|
||||
let tri = *self.t(t)?;
|
||||
for i in 0..3 {
|
||||
let nb = tri.n[i];
|
||||
if nb == NONE || tri.fixed[i] || self.marked(nb) || excluded.binary_search(&nb).is_ok() {
|
||||
continue;
|
||||
}
|
||||
let o = *self.t(nb)?;
|
||||
if incircle(self.p(o.v[0])?, self.p(o.v[1])?, self.p(o.v[2])?, p) == Sign::Positive {
|
||||
set(&mut self.mark, nb);
|
||||
cavity.push(nb);
|
||||
}
|
||||
}
|
||||
}
|
||||
let mut edges = Vec::new();
|
||||
for &t in &cavity {
|
||||
let tri = *self.t(t)?;
|
||||
for i in 0..3 {
|
||||
let nb = tri.n[i];
|
||||
// Shared with another cavity triangle (a constrained one is reported by `insert`).
|
||||
if nb != NONE && self.marked(nb) {
|
||||
continue;
|
||||
}
|
||||
edges.push((tri.v[(i + 1) % 3], tri.v[(i + 2) % 3], nb, tri.fixed[i], t));
|
||||
}
|
||||
}
|
||||
Ok((cavity, edges))
|
||||
}
|
||||
|
||||
fn retriangulate(&mut self, v: u32, cavity: &[u32], edges: &[(u32, u32, u32, bool, u32)], inside: bool) -> Result<()> {
|
||||
for &t in cavity {
|
||||
self.tm(t)?.alive = false;
|
||||
self.free.push(t);
|
||||
}
|
||||
let mut made: Vec<(u32, u32, u32)> = Vec::with_capacity(edges.len()); // (a, b, tri)
|
||||
for &(a, b, nb, fixed, _) in edges {
|
||||
let t = self.alloc(Tri { v: [a, b, v], n: [NONE, NONE, nb], fixed: [false, false, fixed], inside, alive: true })?;
|
||||
made.push((a, b, t));
|
||||
if nb != NONE {
|
||||
self.relink(nb, a, b, t)?;
|
||||
}
|
||||
}
|
||||
let mut by_a: Vec<(u32, u32)> = made.iter().map(|&(a, _, t)| (a, t)).collect();
|
||||
let mut by_b: Vec<(u32, u32)> = made.iter().map(|&(_, b, t)| (b, t)).collect();
|
||||
by_a.sort_unstable();
|
||||
by_b.sort_unstable();
|
||||
let find = |list: &[(u32, u32)], key: u32| -> Result<u32> {
|
||||
let i = list.binary_search_by_key(&key, |e| e.0).map_err(|_| err("cavity boundary is not a closed loop"))?;
|
||||
list.get(i).map(|e| e.1).ok_or_else(|| err("cavity lookup"))
|
||||
};
|
||||
for &(a, b, t) in &made {
|
||||
let next = find(&by_a, b)?; // the triangle (b, c, v): shares edge b–v
|
||||
let prev = find(&by_b, a)?; // the triangle (z, a, v): shares edge v–a
|
||||
let tri = self.tm(t)?;
|
||||
tri.n[0] = next;
|
||||
tri.n[1] = prev;
|
||||
for u in [a, b, v] {
|
||||
if let Some(x) = self.vt.get_mut(u as usize) {
|
||||
*x = t;
|
||||
}
|
||||
}
|
||||
self.last = t;
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// In triangle `t`, point the edge from `b` to `a` at `to`.
|
||||
fn relink(&mut self, t: u32, a: u32, b: u32, to: u32) -> Result<()> {
|
||||
let tri = self.tm(t)?;
|
||||
let i = edge_index(tri, b, a).ok_or_else(|| err("neighbour does not share the edge"))?;
|
||||
tri.n[i] = to;
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Visit the triangles around vertex `v` counterclockwise, each with `v`'s local index,
|
||||
/// until `visit` returns a value (or the rotation closes: `None`).
|
||||
fn around<R>(&self, v: u32, mut visit: impl FnMut(u32, &Tri, usize) -> Result<Option<R>>) -> Result<Option<R>> {
|
||||
let start = self.vt.get(v as usize).copied().ok_or_else(|| err("vertex"))?;
|
||||
let mut t = start;
|
||||
for _ in 0..self.tris.len() + 1 {
|
||||
let tri = self.t(t)?;
|
||||
let k = tri.v.iter().position(|&x| x == v).ok_or_else(|| err("vertex map is stale"))?;
|
||||
if let Some(r) = visit(t, tri, k)? {
|
||||
return Ok(Some(r));
|
||||
}
|
||||
t = tri.n[(k + 1) % 3];
|
||||
if t == NONE || t == start {
|
||||
return Ok(None);
|
||||
}
|
||||
}
|
||||
Err(err("vertex rotation did not close"))
|
||||
}
|
||||
|
||||
/// The triangle and local index of the directed edge `a → b` (the edge opposite `v[i]` runs
|
||||
/// from `v[i+1]` to `v[i+2]`).
|
||||
fn find_edge(&self, a: u32, b: u32) -> Result<Option<(u32, usize)>> {
|
||||
self.around(a, |t, tri, k| Ok((tri.v[(k + 1) % 3] == b).then_some((t, (k + 2) % 3))))
|
||||
}
|
||||
|
||||
/// Flip the edge opposite `v[i]` of `t`. Returns the new edge's end points.
|
||||
fn flip(&mut self, t: u32, i: usize) -> Result<(u32, u32)> {
|
||||
let a = *self.t(t)?;
|
||||
let (p, q, r) = (a.v[i], a.v[(i + 1) % 3], a.v[(i + 2) % 3]);
|
||||
let t2 = a.n[i];
|
||||
let b = *self.t(t2)?;
|
||||
let j = edge_index(&b, r, q).ok_or_else(|| err("flip: neighbour mismatch"))?;
|
||||
let s = b.v[j];
|
||||
let (n_pq, f_pq) = (a.n[(i + 2) % 3], a.fixed[(i + 2) % 3]);
|
||||
let (n_rp, f_rp) = (a.n[(i + 1) % 3], a.fixed[(i + 1) % 3]);
|
||||
let (n_sr, f_sr) = (b.n[(j + 2) % 3], b.fixed[(j + 2) % 3]);
|
||||
let (n_qs, f_qs) = (b.n[(j + 1) % 3], b.fixed[(j + 1) % 3]);
|
||||
*self.tm(t)? = Tri { v: [p, q, s], n: [n_qs, t2, n_pq], fixed: [f_qs, false, f_pq], inside: a.inside, alive: true };
|
||||
*self.tm(t2)? = Tri { v: [p, s, r], n: [n_sr, n_rp, t], fixed: [f_sr, f_rp, false], inside: b.inside, alive: true };
|
||||
if n_qs != NONE {
|
||||
self.relink(n_qs, q, s, t)?;
|
||||
}
|
||||
if n_rp != NONE {
|
||||
self.relink(n_rp, r, p, t2)?;
|
||||
}
|
||||
for (u, x) in [(p, t), (q, t), (s, t2), (r, t2)] {
|
||||
if let Some(y) = self.vt.get_mut(u as usize) {
|
||||
*y = x;
|
||||
}
|
||||
}
|
||||
Ok((p, s))
|
||||
}
|
||||
|
||||
/// The edge opposite `v[i]` of `t` with its far vertex, when it is flippable (strictly convex
|
||||
/// quadrilateral).
|
||||
fn flippable(&self, t: u32, i: usize) -> Result<Option<u32>> {
|
||||
let a = *self.t(t)?;
|
||||
let t2 = a.n[i];
|
||||
if t2 == NONE || a.fixed[i] {
|
||||
return Ok(None);
|
||||
}
|
||||
let (p, q, r) = (a.v[i], a.v[(i + 1) % 3], a.v[(i + 2) % 3]);
|
||||
let b = *self.t(t2)?;
|
||||
let s = b.v[edge_index(&b, r, q).ok_or_else(|| err("neighbour mismatch"))?];
|
||||
let (pp, ps) = (self.p(p)?, self.p(s)?);
|
||||
Ok((orient2d(pp, ps, self.p(q)?) * orient2d(pp, ps, self.p(r)?) == Sign::Negative).then_some(s))
|
||||
}
|
||||
|
||||
/// Make `a–b` an edge (Sloan 1993) and mark it constrained.
|
||||
fn recover(&mut self, a: u32, b: u32) -> Result<()> {
|
||||
if self.find_edge(a, b)?.is_none() {
|
||||
let crossing = self.crossing_edges(a, b)?;
|
||||
let (pa, pb) = (*self.p(a)?, *self.p(b)?);
|
||||
let mut queue: VecDeque<(u32, u32)> = crossing.into_iter().collect();
|
||||
let cap = 1000 + 16 * queue.len() * queue.len();
|
||||
let mut fresh = Vec::new();
|
||||
let mut steps = 0usize;
|
||||
while let Some((u, w)) = queue.pop_front() {
|
||||
steps += 1;
|
||||
if steps > cap {
|
||||
return Err(Error::Limit(format!("boundary edge recovery did not finish in {cap} flips")));
|
||||
}
|
||||
let (t, i) = self.find_edge(u, w)?.ok_or_else(|| err("crossing edge vanished"))?;
|
||||
match self.flippable(t, i)? {
|
||||
Some(_) => {
|
||||
let (x, y) = self.flip(t, i)?;
|
||||
let crosses =
|
||||
x != a && x != b && y != a && y != b && orient2d(&pa, &pb, self.p(x)?) * orient2d(&pa, &pb, self.p(y)?) == Sign::Negative;
|
||||
if crosses { queue.push_back((x, y)) } else { fresh.push((x, y)) }
|
||||
}
|
||||
None => queue.push_back((u, w)),
|
||||
}
|
||||
}
|
||||
self.mark_fixed(a, b)?;
|
||||
self.lawson(fresh)?;
|
||||
return Ok(());
|
||||
}
|
||||
self.mark_fixed(a, b)
|
||||
}
|
||||
|
||||
fn mark_fixed(&mut self, a: u32, b: u32) -> Result<()> {
|
||||
let (t, i) = self.find_edge(a, b)?.ok_or_else(|| err("recovered edge is missing"))?;
|
||||
let nb = self.t(t)?.n[i];
|
||||
self.tm(t)?.fixed[i] = true;
|
||||
if nb != NONE {
|
||||
let tri = self.tm(nb)?;
|
||||
if let Some(k) = edge_index(tri, b, a) {
|
||||
tri.fixed[k] = true;
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// The edges crossed by the open segment `a–b`, from `a` to `b`.
|
||||
fn crossing_edges(&self, a: u32, b: u32) -> Result<Vec<(u32, u32)>> {
|
||||
let (pa, pb) = (*self.p(a)?, *self.p(b)?);
|
||||
let found = self.around(a, |t, tri, k| {
|
||||
let (u, w) = (tri.v[(k + 1) % 3], tri.v[(k + 2) % 3]);
|
||||
let (ou, ow) = (orient2d(&pa, self.p(u)?, &pb), orient2d(&pa, self.p(w)?, &pb));
|
||||
if ou.is_zero() || ow.is_zero() {
|
||||
let x = if ou.is_zero() { u } else { w };
|
||||
let px = self.xy(x)?;
|
||||
let (qa, qb) = (pa.coords(), pb.coords());
|
||||
if (px[0] - qa[0]) * (qb[0] - qa[0]) + (px[1] - qa[1]) * (qb[1] - qa[1]) > 0.0 {
|
||||
return Err(Error::Invalid("a point lies on a boundary edge".into()));
|
||||
}
|
||||
}
|
||||
Ok((ou == Sign::Positive && ow == Sign::Negative).then_some((t, u, w)))
|
||||
})?;
|
||||
let (mut t, mut right, mut left) = found.ok_or_else(|| err("segment start not found"))?;
|
||||
let mut out = vec![(right, left)];
|
||||
for _ in 0..self.tris.len() + 1 {
|
||||
let tri = *self.t(t)?;
|
||||
let i = tri.v.iter().position(|&x| x != right && x != left).ok_or_else(|| err("walk"))?;
|
||||
let next = tri.n[i];
|
||||
let nt = *self.t(next)?;
|
||||
let d = *nt.v.iter().find(|&&x| x != right && x != left).ok_or_else(|| err("walk"))?;
|
||||
if d == b {
|
||||
return Ok(out);
|
||||
}
|
||||
match orient2d(&pa, &pb, self.p(d)?) {
|
||||
Sign::Positive => left = d,
|
||||
Sign::Negative => right = d,
|
||||
Sign::Zero => return Err(Error::Invalid("a point lies on a boundary edge".into())),
|
||||
}
|
||||
out.push((right, left));
|
||||
t = next;
|
||||
}
|
||||
Err(err("segment walk did not reach its end"))
|
||||
}
|
||||
|
||||
/// Lawson flips from the given edges until every unconstrained edge is locally Delaunay.
|
||||
fn lawson(&mut self, edges: Vec<(u32, u32)>) -> Result<()> {
|
||||
let mut stack = edges;
|
||||
let cap = 1000 + 64 * self.tris.len();
|
||||
let mut steps = 0usize;
|
||||
while let Some((u, w)) = stack.pop() {
|
||||
steps += 1;
|
||||
if steps > cap {
|
||||
return Err(Error::Limit("Delaunay restoration did not finish".into()));
|
||||
}
|
||||
let Some((t, i)) = self.find_edge(u, w)? else { continue };
|
||||
let tri = *self.t(t)?;
|
||||
if tri.fixed[i] || tri.n[i] == NONE {
|
||||
continue;
|
||||
}
|
||||
let Some(s) = self.flippable(t, i)? else { continue };
|
||||
let [a, b, c] = tri.v;
|
||||
if incircle(self.p(a)?, self.p(b)?, self.p(c)?, self.p(s)?) != Sign::Positive {
|
||||
continue;
|
||||
}
|
||||
let (p, _) = self.flip(t, i)?;
|
||||
let (q, r) = (tri.v[(i + 1) % 3], tri.v[(i + 2) % 3]);
|
||||
stack.extend([(p, q), (q, s), (s, r), (r, p)]);
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Mark the triangles inside the region: those separated from the super-triangle by an odd
|
||||
/// number of constrained edges.
|
||||
fn classify(&mut self) -> Result<()> {
|
||||
let n = self.tris.len();
|
||||
let mut depth = vec![u32::MAX; n];
|
||||
let start = self.vt.get(self.pts.len().saturating_sub(1)).copied().ok_or_else(|| err("super vertex"))?;
|
||||
let mut current = vec![start];
|
||||
if let Some(d) = depth.get_mut(start as usize) {
|
||||
*d = 0;
|
||||
}
|
||||
let mut level = 0u32;
|
||||
while !current.is_empty() {
|
||||
let mut next = Vec::new();
|
||||
let mut k = 0;
|
||||
while let Some(&t) = current.get(k) {
|
||||
k += 1;
|
||||
let tri = *self.t(t)?;
|
||||
for i in 0..3 {
|
||||
let nb = tri.n[i];
|
||||
if nb == NONE || depth.get(nb as usize).is_none_or(|&d| d != u32::MAX) {
|
||||
continue;
|
||||
}
|
||||
if tri.fixed[i] {
|
||||
next.push(nb);
|
||||
} else {
|
||||
if let Some(d) = depth.get_mut(nb as usize) {
|
||||
*d = level;
|
||||
}
|
||||
current.push(nb);
|
||||
}
|
||||
}
|
||||
}
|
||||
level += 1;
|
||||
next.retain(|&t| depth.get(t as usize).is_some_and(|&d| d == u32::MAX));
|
||||
next.sort_unstable();
|
||||
next.dedup();
|
||||
for &t in &next {
|
||||
if let Some(d) = depth.get_mut(t as usize) {
|
||||
*d = level;
|
||||
}
|
||||
}
|
||||
current = next;
|
||||
if level > 4 {
|
||||
break;
|
||||
}
|
||||
}
|
||||
for (tri, d) in self.tris.iter_mut().zip(depth) {
|
||||
tri.inside = tri.alive && d == 1;
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Try to insert Steiner point `p` (located from `hint`), at least `min_distance` from the
|
||||
/// vertices of its cavity.
|
||||
fn try_steiner(&mut self, p: Point2, hint: u32, min_distance: f64) -> Result<bool> {
|
||||
let v = self.pts.len() as u32;
|
||||
self.pts.push(p.coords());
|
||||
self.p2.push(p);
|
||||
self.vt.push(NONE);
|
||||
self.last = hint;
|
||||
let ok = self.insert(v, Some(min_distance))?;
|
||||
if !ok {
|
||||
self.pts.pop();
|
||||
self.p2.pop();
|
||||
self.vt.pop();
|
||||
}
|
||||
Ok(ok)
|
||||
}
|
||||
|
||||
/// Delaunay refinement of the inside triangles to `scale × size field`. A Steiner point `p`
|
||||
/// for local size `h` is skipped when `allows(p, h)` refuses it (the caller's clearance from
|
||||
/// the true boundary) or when it comes closer than [`CLEARANCE`]` × h` to a vertex.
|
||||
pub fn refine(&mut self, size: &dyn SizeField, scale: f64, allows: &dyn Fn(&Point2, f64) -> bool) -> Result<()> {
|
||||
for _ in 0..MAX_REFINE_PASSES {
|
||||
let mut inserted = 0usize;
|
||||
let n = self.tris.len();
|
||||
for t in 0..n as u32 {
|
||||
let tri = *self.t(t)?;
|
||||
if !tri.alive || !tri.inside {
|
||||
continue;
|
||||
}
|
||||
let [a, b, c] = [self.xy(tri.v[0])?, self.xy(tri.v[1])?, self.xy(tri.v[2])?];
|
||||
let centroid = [(a[0] + b[0] + c[0]) / 3.0, (a[1] + b[1] + c[1]) / 3.0];
|
||||
let h = scale * checked_size(size, [centroid[0], centroid[1], 0.0])?;
|
||||
let (bx, by, cx, cy) = (b[0] - a[0], b[1] - a[1], c[0] - a[0], c[1] - a[1]);
|
||||
let d = 2.0 * (bx * cy - by * cx);
|
||||
let (b2, c2) = (bx * bx + by * by, cx * cx + cy * cy);
|
||||
let (ux, uy) = ((cy * b2 - by * c2) / d, (bx * c2 - cx * b2) / d);
|
||||
let r = (ux * ux + uy * uy).sqrt();
|
||||
let shortest = b2.min(c2).min((cx - bx).powi(2) + (cy - by).powi(2)).sqrt();
|
||||
if !r.is_finite() || !(d > 0.0) {
|
||||
continue;
|
||||
}
|
||||
let too_big = r > 0.75 * h;
|
||||
let poor = r > std::f64::consts::SQRT_2 * shortest;
|
||||
if !too_big && !poor {
|
||||
continue;
|
||||
}
|
||||
let candidates = [Some([a[0] + ux, a[1] + uy]), too_big.then_some(centroid)];
|
||||
for c in candidates.into_iter().flatten() {
|
||||
if let Some(p) = steiner_point(c).filter(|p| allows(p, h))
|
||||
&& self.try_steiner(p, t, CLEARANCE * h)?
|
||||
{
|
||||
inserted += 1;
|
||||
break;
|
||||
}
|
||||
}
|
||||
}
|
||||
if inserted == 0 {
|
||||
break;
|
||||
}
|
||||
}
|
||||
#[cfg(test)]
|
||||
assert_eq!(self.verify(), Ok(()), "after refinement");
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// The inside triangles around vertex `v`.
|
||||
fn inside_fan(&self, v: u32) -> Result<Vec<u32>> {
|
||||
let mut fan = Vec::new();
|
||||
self.around(v, |t, tri, _| {
|
||||
if tri.inside {
|
||||
fan.push(t);
|
||||
}
|
||||
Ok(None::<()>)
|
||||
})?;
|
||||
Ok(fan)
|
||||
}
|
||||
|
||||
/// [`Cdt::ensure_fan`] for every vertex with its count in `need`.
|
||||
pub fn ensure_fans(&mut self, need: &[usize]) -> Result<()> {
|
||||
for (v, &k) in need.iter().enumerate() {
|
||||
if k > 1 {
|
||||
self.ensure_fan(v as u32, k)?;
|
||||
}
|
||||
}
|
||||
#[cfg(test)]
|
||||
assert_eq!(self.verify(), Ok(()), "after fans");
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Give boundary vertex `v` at least `count` inside triangles by inserting the centroid of
|
||||
/// its widest triangle (bounded attempts; the caller's validity gate judges the result).
|
||||
fn ensure_fan(&mut self, v: u32, count: usize) -> Result<()> {
|
||||
for _ in 0..count + 2 {
|
||||
let fan = self.inside_fan(v)?;
|
||||
if fan.len() >= count {
|
||||
return Ok(());
|
||||
}
|
||||
let pv = self.xy(v)?;
|
||||
let mut widest: Option<(f64, u32, [f64; 2])> = None;
|
||||
for &t in &fan {
|
||||
let tri = *self.t(t)?;
|
||||
let k = tri.v.iter().position(|&x| x == v).ok_or_else(|| err("fan"))?;
|
||||
let (a, b) = (self.xy(tri.v[(k + 1) % 3])?, self.xy(tri.v[(k + 2) % 3])?);
|
||||
let (u, w) = ([a[0] - pv[0], a[1] - pv[1]], [b[0] - pv[0], b[1] - pv[1]]);
|
||||
let cos = (u[0] * w[0] + u[1] * w[1]) / ((u[0] * u[0] + u[1] * u[1]).sqrt() * (w[0] * w[0] + w[1] * w[1]).sqrt());
|
||||
let c = [(pv[0] + a[0] + b[0]) / 3.0, (pv[1] + a[1] + b[1]) / 3.0];
|
||||
if widest.is_none_or(|(m, _, _)| cos < m) {
|
||||
widest = Some((cos, t, c));
|
||||
}
|
||||
}
|
||||
let Some((t, p)) = widest.and_then(|(_, t, c)| steiner_point(c).map(|p| (t, p))) else { return Ok(()) };
|
||||
if !self.try_steiner(p, t, 0.0)? {
|
||||
return Ok(());
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Consistency check for tests: every live triangle counterclockwise (exact), neighbours
|
||||
/// mutual, constrained flags matching on both sides.
|
||||
#[cfg(test)]
|
||||
pub fn verify(&self) -> std::result::Result<(), String> {
|
||||
for (t, tri) in self.tris.iter().enumerate() {
|
||||
if !tri.alive {
|
||||
continue;
|
||||
}
|
||||
let [a, b, c] = tri.v;
|
||||
let o = orient2d(&self.p2[a as usize], &self.p2[b as usize], &self.p2[c as usize]);
|
||||
if o != Sign::Positive {
|
||||
return Err(format!("triangle {t} {:?} has orientation {o:?} (inside {})", tri.v, tri.inside));
|
||||
}
|
||||
for i in 0..3 {
|
||||
let nb = tri.n[i];
|
||||
if nb == NONE {
|
||||
continue;
|
||||
}
|
||||
let o = &self.tris[nb as usize];
|
||||
let (p, q) = (tri.v[(i + 1) % 3], tri.v[(i + 2) % 3]);
|
||||
let back = (0..3).find(|&j| o.v[(j + 1) % 3] == q && o.v[(j + 2) % 3] == p);
|
||||
match back {
|
||||
Some(j) if o.n[j] == t as u32 && o.alive && o.fixed[j] == tri.fixed[i] => {
|
||||
let s = o.v[j];
|
||||
if !tri.fixed[i]
|
||||
&& incircle(&self.p2[a as usize], &self.p2[b as usize], &self.p2[c as usize], &self.p2[s as usize]) == Sign::Positive
|
||||
{
|
||||
return Err(format!("triangle {t} edge {i} is not locally Delaunay (inside {})", tri.inside));
|
||||
}
|
||||
}
|
||||
_ => return Err(format!("triangle {t} edge {i}: neighbour {nb} does not match")),
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// The inside triangles.
|
||||
pub fn into_triangulation(self) -> Triangulation {
|
||||
let tris = self.tris.iter().filter(|t| t.alive && t.inside).map(|t| t.v).collect();
|
||||
Triangulation { pts: self.pts, p2: self.p2, tris }
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,309 @@
|
||||
//! Validation of a planar region given as boundary loops: finite, admissible coordinates, no
|
||||
//! zero-length edges, no repeated points, no crossing or touching edges, holes inside the outer
|
||||
//! loop and not inside each other. Every geometric decision is an exact `orient2d` or an exact
|
||||
//! coordinate comparison.
|
||||
|
||||
use std::cmp::Ordering;
|
||||
|
||||
use crate::predicates::{Point2, Sign, orient2d, snap_to_domain};
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// The most boundary points of one region.
|
||||
pub const MAX_BOUNDARY_POINTS: usize = 1 << 20;
|
||||
/// The most loops (outer plus holes) of one region.
|
||||
pub const MAX_LOOPS: usize = 1 << 14;
|
||||
/// The most segment-cell insertions and pair tests of the intersection check.
|
||||
const MAX_CHECK_WORK: usize = 1 << 28;
|
||||
|
||||
/// A validated region: the points in input order and the loops as point ids, the outer loop
|
||||
/// first and counterclockwise, holes clockwise.
|
||||
#[derive(Debug, Clone)]
|
||||
pub(crate) struct Region {
|
||||
pub pts: Vec<[f64; 2]>,
|
||||
pub p2: Vec<Point2>,
|
||||
pub loops: Vec<Vec<u32>>,
|
||||
}
|
||||
|
||||
impl Region {
|
||||
pub fn point(&self, v: u32) -> Result<&Point2> {
|
||||
self.p2.get(v as usize).ok_or_else(|| Error::Failed(format!("point {v} out of range")))
|
||||
}
|
||||
}
|
||||
|
||||
/// The bounding box `(lo, hi)` of points (`lo > hi` when there are none).
|
||||
pub(crate) fn bbox(pts: impl IntoIterator<Item = [f64; 2]>) -> ([f64; 2], [f64; 2]) {
|
||||
pts.into_iter()
|
||||
.fold(([f64::INFINITY; 2], [f64::NEG_INFINITY; 2]), |(lo, hi), p| ([lo[0].min(p[0]), lo[1].min(p[1])], [hi[0].max(p[0]), hi[1].max(p[1])]))
|
||||
}
|
||||
|
||||
/// Admissible exact-predicate point from user coordinates (finite, within the model limit; tiny
|
||||
/// magnitudes rounded to zero).
|
||||
pub(crate) fn admissible(c: [f64; 2]) -> Result<[f64; 2]> {
|
||||
modelcraft_model::check_coordinates(&c)?;
|
||||
Ok([snap_to_domain(c[0]), snap_to_domain(c[1])])
|
||||
}
|
||||
|
||||
/// Validate loops given as point lists (the first is the outer boundary, the rest are holes, in
|
||||
/// either orientation). Point ids number the points in input order, loop by loop.
|
||||
pub(crate) fn region(input: &[Vec<[f64; 2]>]) -> Result<Region> {
|
||||
if input.is_empty() {
|
||||
return Err(Error::Invalid("a region needs an outer boundary loop".into()));
|
||||
}
|
||||
if input.len() > MAX_LOOPS {
|
||||
return Err(Error::Limit(format!("{} loops exceed the limit of {MAX_LOOPS}", input.len())));
|
||||
}
|
||||
let total: usize = input.iter().map(Vec::len).sum();
|
||||
if total > MAX_BOUNDARY_POINTS {
|
||||
return Err(Error::Limit(format!("{total} boundary points exceed the limit of {MAX_BOUNDARY_POINTS}")));
|
||||
}
|
||||
let mut pts = Vec::with_capacity(total);
|
||||
let mut loops = Vec::with_capacity(input.len());
|
||||
for (k, l) in input.iter().enumerate() {
|
||||
if l.len() < 3 {
|
||||
return Err(Error::Invalid(format!("loop {k} has {} points; a loop needs at least 3", l.len())));
|
||||
}
|
||||
let mut ids = Vec::with_capacity(l.len());
|
||||
for (i, c) in l.iter().enumerate() {
|
||||
let c = admissible(*c).map_err(|e| Error::Invalid(format!("loop {k} point {i}: {e}")))?;
|
||||
ids.push(pts.len() as u32);
|
||||
pts.push(c);
|
||||
}
|
||||
loops.push(ids);
|
||||
}
|
||||
let p2 = pts.iter().map(|c| Point2::new(c[0], c[1])).collect::<Result<Vec<_>>>()?;
|
||||
let mut r = Region { pts, p2, loops };
|
||||
check_distinct(&r)?;
|
||||
check_simple(&r)?;
|
||||
orient(&mut r)?;
|
||||
check_nesting(&r)?;
|
||||
Ok(r)
|
||||
}
|
||||
|
||||
fn lex(a: &[f64; 2], b: &[f64; 2]) -> Ordering {
|
||||
a[0].total_cmp(&b[0]).then(a[1].total_cmp(&b[1]))
|
||||
}
|
||||
|
||||
/// Report a point by its loop and position.
|
||||
fn locate(r: &Region, v: u32) -> String {
|
||||
for (k, l) in r.loops.iter().enumerate() {
|
||||
if let Some(i) = l.iter().position(|&x| x == v) {
|
||||
return format!("loop {k} point {i}");
|
||||
}
|
||||
}
|
||||
format!("point {v}")
|
||||
}
|
||||
|
||||
/// No point appears twice (this also rejects zero-length edges).
|
||||
fn check_distinct(r: &Region) -> Result<()> {
|
||||
let mut order: Vec<u32> = (0..r.pts.len() as u32).collect();
|
||||
order.sort_by(|&a, &b| match (r.pts.get(a as usize), r.pts.get(b as usize)) {
|
||||
(Some(p), Some(q)) => lex(p, q).then(a.cmp(&b)),
|
||||
_ => a.cmp(&b),
|
||||
});
|
||||
for w in order.windows(2) {
|
||||
if let [a, b] = *w
|
||||
&& r.pts.get(a as usize) == r.pts.get(b as usize)
|
||||
{
|
||||
let what = if r.loops.iter().any(|l| l.windows(2).any(|e| e == [a, b] || e == [b, a]) || is_closing(l, a, b)) {
|
||||
"zero-length edge"
|
||||
} else {
|
||||
"repeated point"
|
||||
};
|
||||
return Err(Error::Invalid(format!("{what}: {} and {} coincide at {:?}", locate(r, a), locate(r, b), r.pts.get(a as usize))));
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
fn is_closing(l: &[u32], a: u32, b: u32) -> bool {
|
||||
matches!((l.first(), l.last()), (Some(&f), Some(&z)) if (f == a && z == b) || (f == b && z == a))
|
||||
}
|
||||
|
||||
/// The boundary segments `(a, b)` of every loop, in loop order.
|
||||
pub(crate) fn segments(loops: &[Vec<u32>]) -> Vec<(u32, u32)> {
|
||||
let mut s = Vec::new();
|
||||
for l in loops {
|
||||
for (i, &a) in l.iter().enumerate() {
|
||||
if let Some(&b) = l.get((i + 1) % l.len()) {
|
||||
s.push((a, b));
|
||||
}
|
||||
}
|
||||
}
|
||||
s
|
||||
}
|
||||
|
||||
/// `c` lies within the bounding box of `a`, `b` (for collinear points: on the segment).
|
||||
fn on_segment(a: [f64; 2], b: [f64; 2], c: [f64; 2]) -> bool {
|
||||
a[0].min(b[0]) <= c[0] && c[0] <= a[0].max(b[0]) && a[1].min(b[1]) <= c[1] && c[1] <= a[1].max(b[1])
|
||||
}
|
||||
|
||||
/// Whether two boundary segments conflict: they cross or touch, or (when they share an end
|
||||
/// point) they fold back on each other.
|
||||
fn segments_conflict(r: &Region, s: (u32, u32), t: (u32, u32)) -> Result<bool> {
|
||||
let (a, b, c, d) = (r.point(s.0)?, r.point(s.1)?, r.point(t.0)?, r.point(t.1)?);
|
||||
let shared = [(s.0 == t.0, b, d, a), (s.0 == t.1, b, c, a), (s.1 == t.0, a, d, b), (s.1 == t.1, a, c, b)];
|
||||
if let Some(&(_, x, y, o)) = shared.iter().find(|e| e.0) {
|
||||
// Segments o–x and o–y: a conflict when collinear and on the same side of o.
|
||||
if orient2d(o, x, y) != Sign::Zero {
|
||||
return Ok(false);
|
||||
}
|
||||
let (o, x, y) = (o.coords(), x.coords(), y.coords());
|
||||
let dot = (x[0] - o[0]) * (y[0] - o[0]) + (x[1] - o[1]) * (y[1] - o[1]);
|
||||
return Ok(dot > 0.0);
|
||||
}
|
||||
let (o1, o2, o3, o4) = (orient2d(a, b, c), orient2d(a, b, d), orient2d(c, d, a), orient2d(c, d, b));
|
||||
let proper = (o1 * o2) == Sign::Negative && (o3 * o4) == Sign::Negative;
|
||||
let (pa, pb, pc, pd) = (a.coords(), b.coords(), c.coords(), d.coords());
|
||||
Ok(proper
|
||||
|| (o1.is_zero() && on_segment(pa, pb, pc))
|
||||
|| (o2.is_zero() && on_segment(pa, pb, pd))
|
||||
|| (o3.is_zero() && on_segment(pc, pd, pa))
|
||||
|| (o4.is_zero() && on_segment(pc, pd, pb)))
|
||||
}
|
||||
|
||||
/// No two boundary segments cross, touch or overlap. Segments are bucketed on a uniform grid
|
||||
/// over the bounding box; each pair is tested once, in the first cell both bounding boxes cover.
|
||||
pub(crate) fn check_simple(r: &Region) -> Result<()> {
|
||||
let segs = segments(&r.loops);
|
||||
let n = segs.len();
|
||||
let (lo, hi) = bbox(r.pts.iter().copied());
|
||||
let g = ((n as f64).sqrt().ceil() as usize).clamp(1, 1024);
|
||||
let cell = |x: f64, axis: usize| -> usize {
|
||||
let (l, h) = (lo[axis], hi[axis]);
|
||||
if !(h > l) {
|
||||
return 0;
|
||||
}
|
||||
(((x - l) / (h - l) * g as f64) as usize).min(g - 1)
|
||||
};
|
||||
let mut boxes = Vec::with_capacity(n);
|
||||
let mut grid: Vec<Vec<u32>> = vec![Vec::new(); g * g];
|
||||
let mut work = 0usize;
|
||||
for (i, &(a, b)) in segs.iter().enumerate() {
|
||||
let (pa, pb) = (r.point(a)?.coords(), r.point(b)?.coords());
|
||||
let bx = [cell(pa[0].min(pb[0]), 0), cell(pa[1].min(pb[1]), 1), cell(pa[0].max(pb[0]), 0), cell(pa[1].max(pb[1]), 1)];
|
||||
for cy in bx[1]..=bx[3] {
|
||||
for cx in bx[0]..=bx[2] {
|
||||
work += 1;
|
||||
if let Some(c) = grid.get_mut(cy * g + cx) {
|
||||
c.push(i as u32);
|
||||
}
|
||||
}
|
||||
}
|
||||
if work > MAX_CHECK_WORK {
|
||||
return Err(Error::Limit("boundary intersection check exceeded its work limit".into()));
|
||||
}
|
||||
boxes.push(bx);
|
||||
}
|
||||
for (ci, members) in grid.iter().enumerate() {
|
||||
let (cx, cy) = (ci % g, ci / g);
|
||||
for (k, &i) in members.iter().enumerate() {
|
||||
for &j in members.get(k + 1..).unwrap_or(&[]) {
|
||||
let (Some(bi), Some(bj), Some(&si), Some(&sj)) =
|
||||
(boxes.get(i as usize), boxes.get(j as usize), segs.get(i as usize), segs.get(j as usize))
|
||||
else {
|
||||
continue;
|
||||
};
|
||||
if bi[0].max(bj[0]) != cx || bi[1].max(bj[1]) != cy {
|
||||
continue;
|
||||
}
|
||||
work += 1;
|
||||
if work > MAX_CHECK_WORK {
|
||||
return Err(Error::Limit("boundary intersection check exceeded its work limit".into()));
|
||||
}
|
||||
if segments_conflict(r, si, sj)? {
|
||||
return Err(Error::Invalid(format!(
|
||||
"boundary edges cross, touch or overlap: {}–{} and {}–{}",
|
||||
locate(r, si.0),
|
||||
locate(r, si.1),
|
||||
locate(r, sj.0),
|
||||
locate(r, sj.1)
|
||||
)));
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// The exact orientation of a simple loop: the sign of the turn at its lexicographically
|
||||
/// smallest point, a vertex of its convex hull.
|
||||
pub(crate) fn loop_orientation(r: &Region, l: &[u32]) -> Result<Sign> {
|
||||
let (i, _) = l
|
||||
.iter()
|
||||
.enumerate()
|
||||
.filter_map(|(i, &v)| r.pts.get(v as usize).map(|p| (i, p)))
|
||||
.min_by(|x, y| lex(x.1, y.1))
|
||||
.ok_or_else(|| Error::Invalid("empty loop".into()))?;
|
||||
let n = l.len();
|
||||
let at = |k: usize| l.get(k % n).copied().ok_or_else(|| Error::Failed("loop index".into()));
|
||||
Ok(orient2d(r.point(at(i + n - 1)?)?, r.point(at(i)?)?, r.point(at(i + 1)?)?))
|
||||
}
|
||||
|
||||
/// The outer loop is counterclockwise and the holes clockwise (no repair).
|
||||
pub(crate) fn check_orientation(r: &Region) -> Result<()> {
|
||||
for (k, l) in r.loops.iter().enumerate() {
|
||||
let want = if k == 0 { Sign::Positive } else { Sign::Negative };
|
||||
if loop_orientation(r, l)? != want {
|
||||
return Err(Error::Invalid(format!("loop {k} has the wrong orientation")));
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Outer loop counterclockwise, holes clockwise.
|
||||
fn orient(r: &mut Region) -> Result<()> {
|
||||
for k in 0..r.loops.len() {
|
||||
let l = r.loops.get(k).cloned().unwrap_or_default();
|
||||
let want = if k == 0 { Sign::Positive } else { Sign::Negative };
|
||||
let s = loop_orientation(r, &l)?;
|
||||
if s == Sign::Zero {
|
||||
return Err(Error::Invalid(format!("loop {k} is degenerate (all points collinear)")));
|
||||
}
|
||||
if s != want
|
||||
&& let Some(l) = r.loops.get_mut(k)
|
||||
{
|
||||
l.reverse();
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Winding number of loop `l` around `p` (exact: half-open crossing rule with `orient2d`).
|
||||
pub(crate) fn winding(r: &Region, l: &[u32], p: &Point2) -> Result<i64> {
|
||||
let y = p.coords()[1];
|
||||
let mut w = 0i64;
|
||||
for (i, &a) in l.iter().enumerate() {
|
||||
let b = l.get((i + 1) % l.len()).copied().unwrap_or(a);
|
||||
let (pa, pb) = (r.point(a)?, r.point(b)?);
|
||||
let (ya, yb) = (pa.coords()[1], pb.coords()[1]);
|
||||
if ya <= y && yb > y && orient2d(pa, pb, p) == Sign::Positive {
|
||||
w += 1;
|
||||
} else if yb <= y && ya > y && orient2d(pa, pb, p) == Sign::Negative {
|
||||
w -= 1;
|
||||
}
|
||||
}
|
||||
Ok(w)
|
||||
}
|
||||
|
||||
/// Every hole lies inside the outer loop and outside every other hole. The loops do not
|
||||
/// intersect, so one point per hole decides.
|
||||
pub(crate) fn check_nesting(r: &Region) -> Result<()> {
|
||||
let holes = r.loops.len().saturating_sub(1);
|
||||
if holes.saturating_mul(r.pts.len()) > MAX_CHECK_WORK {
|
||||
return Err(Error::Limit(format!("{holes} holes on {} boundary points exceed the nesting-check limit", r.pts.len())));
|
||||
}
|
||||
let outer = r.loops.first().cloned().unwrap_or_default();
|
||||
for (k, hole) in r.loops.iter().enumerate().skip(1) {
|
||||
let first = hole.first().copied().ok_or_else(|| Error::Invalid(format!("loop {k} is empty")))?;
|
||||
let p = r.point(first)?;
|
||||
if winding(r, &outer, p)? == 0 {
|
||||
return Err(Error::Invalid(format!("hole loop {k} lies outside the outer loop")));
|
||||
}
|
||||
for (m, other) in r.loops.iter().enumerate().skip(1) {
|
||||
if m != k && winding(r, other, p)? != 0 {
|
||||
return Err(Error::Invalid(format!("hole loop {k} lies inside hole loop {m}")));
|
||||
}
|
||||
}
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
@@ -0,0 +1,361 @@
|
||||
//! The free all-quadrilateral mesher: half-resolution constrained Delaunay triangulation,
|
||||
//! recombination into quadrilaterals, one uniform split that lands on the prescribed boundary
|
||||
//! nodes, then constrained smoothing. See the module docs of [`super`] for the method and its
|
||||
//! sources.
|
||||
|
||||
use std::collections::BTreeMap;
|
||||
|
||||
use super::cdt::{CLEARANCE, Cdt, Triangulation};
|
||||
use super::domain::{self, Region};
|
||||
use super::guard::Guard;
|
||||
use super::{QuadMesh, QuadOptions, finish};
|
||||
use crate::predicates::{Sign, orient2d};
|
||||
use crate::quality::{planar_corner, turns_left};
|
||||
use crate::size::SizeField;
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// Mesh the region bounded by `loops` (the first is the outer boundary, the rest are holes;
|
||||
/// either orientation) with quadrilaterals only. See [`super::free_quad_mesh`].
|
||||
pub(crate) fn mesh(loops: &[Vec<[f64; 2]>], size: &dyn SizeField, opts: &QuadOptions) -> Result<QuadMesh> {
|
||||
let region = domain::region(loops)?;
|
||||
for (k, l) in region.loops.iter().enumerate() {
|
||||
if l.len() % 2 == 1 {
|
||||
return Err(Error::Invalid(format!(
|
||||
"loop {k} has an odd number ({}) of boundary points; an all-quadrilateral mesh needs an even number on every loop \
|
||||
(choose the edge divisions with modelcraft_mesh::size::loop_divisions or even_divisions)",
|
||||
l.len()
|
||||
)));
|
||||
}
|
||||
}
|
||||
if region.loops.len() == 1 && region.loops.first().is_some_and(|l| l.len() == 4) {
|
||||
let l = region.loops.first().cloned().unwrap_or_default();
|
||||
let quad = [l.first(), l.get(1), l.get(2), l.get(3)];
|
||||
let quad = match quad {
|
||||
[Some(&a), Some(&b), Some(&c), Some(&d)] => [a, b, c, d],
|
||||
_ => return Err(Error::Failed("quad loop".into())),
|
||||
};
|
||||
return finish(region.pts.clone(), vec![quad], region.pts.len(), opts, Default::default());
|
||||
}
|
||||
if let Some((k, l)) = region.loops.iter().enumerate().find(|(_, l)| l.len() < 6) {
|
||||
return Err(Error::Invalid(format!(
|
||||
"loop {k} has {} boundary points; the free quad mesher needs at least 6 on every loop of a region with more than one loop \
|
||||
(split its sides)",
|
||||
l.len()
|
||||
)));
|
||||
}
|
||||
|
||||
// Both parity classes, preferred first; the second is tried when the first cannot be made
|
||||
// valid.
|
||||
let mut last = None;
|
||||
for coarse in coarse_regions(®ion)? {
|
||||
match attempt(®ion, coarse, size, opts) {
|
||||
Ok(mesh) => return Ok(mesh),
|
||||
Err(e @ Error::Failed(_)) => last = Some(e),
|
||||
Err(e) => return Err(e),
|
||||
}
|
||||
}
|
||||
Err(last.unwrap_or_else(|| Error::Failed("no half-resolution boundary".into())))
|
||||
}
|
||||
|
||||
/// One run of the pipeline on one half-resolution region.
|
||||
fn attempt(region: &Region, coarse: Coarse, size: &dyn SizeField, opts: &QuadOptions) -> Result<QuadMesh> {
|
||||
let Coarse { region: coarse, to_input, skipped, before } = coarse;
|
||||
let chords = chords(&coarse, &skipped)?;
|
||||
// For each coarse boundary edge (coarse ids, smaller first), the input point it skips.
|
||||
let boundary_mid: BTreeMap<(u32, u32), u32> = chords.iter().map(|&(a, b, m)| ((a.min(b), a.max(b)), m)).collect();
|
||||
let guard =
|
||||
Guard::new(&chords.iter().map(|&(a, b, m)| Ok((*coarse.point(a)?, *coarse.point(b)?, *region.point(m)?))).collect::<Result<Vec<_>>>()?)?;
|
||||
let need = fan_needs(region, &to_input, &before, &skipped)?;
|
||||
let max_tris = opts.max_elements / 2 + 4 * region.pts.len() + 16;
|
||||
let mut cdt = Cdt::build(&coarse, max_tris)?;
|
||||
cdt.refine(size, 2.0, &|p, h| guard.allows(p, CLEARANCE * h))?;
|
||||
cdt.ensure_fans(&need)?;
|
||||
let tri = cdt.into_triangulation();
|
||||
let (quads, leftovers) = recombine(&tri, &need)?;
|
||||
let count = 4 * quads.len() + 3 * leftovers.len();
|
||||
if count > opts.max_elements {
|
||||
return Err(Error::Limit(format!("the mesh would have {count} quadrilaterals; the limit is {}", opts.max_elements)));
|
||||
}
|
||||
let (nodes, elements) = split(region, &tri, &to_input, &boundary_mid, &quads, &leftovers)?;
|
||||
let stats = super::QuadStats { recombined: quads.len(), leftover_triangles: leftovers.len(), ..Default::default() };
|
||||
finish(nodes, elements, region.pts.len(), opts, stats)
|
||||
}
|
||||
|
||||
/// `1 − cos θ` of the turn at each loop position (0 on a straight run, 2 on a full reversal).
|
||||
fn turns(region: &Region, l: &[u32]) -> Result<Vec<f64>> {
|
||||
let n = l.len();
|
||||
let at = |i: usize| -> Result<[f64; 2]> {
|
||||
let v = l.get(i % n).copied().ok_or_else(|| Error::Failed("loop index".into()))?;
|
||||
region.pts.get(v as usize).copied().ok_or_else(|| Error::Failed("point index".into()))
|
||||
};
|
||||
(0..n)
|
||||
.map(|i| {
|
||||
let (a, b, c) = (at(i + n - 1)?, at(i)?, at(i + 1)?);
|
||||
let (u, w) = ([b[0] - a[0], b[1] - a[1]], [c[0] - b[0], c[1] - b[1]]);
|
||||
let den = (u[0] * u[0] + u[1] * u[1]).sqrt() * (w[0] * w[0] + w[1] * w[1]).sqrt();
|
||||
let cos = (u[0] * w[0] + u[1] * w[1]) / den;
|
||||
Ok(if den > 0.0 && cos.is_finite() { 1.0 - cos } else { 0.0 })
|
||||
})
|
||||
.collect()
|
||||
}
|
||||
|
||||
/// The half-resolution regions to try: every other point of each loop, first keeping the parity
|
||||
/// class with the sharper turns (so loop corners survive when they sit at an even spacing), then
|
||||
/// the other class of every loop. Choices whose chords cross or flip are dropped.
|
||||
fn coarse_regions(region: &Region) -> Result<Vec<Coarse>> {
|
||||
let mut preferred = Vec::with_capacity(region.loops.len());
|
||||
for l in ®ion.loops {
|
||||
let t = turns(region, l)?;
|
||||
let even: f64 = t.iter().step_by(2).sum();
|
||||
let odd: f64 = t.iter().skip(1).step_by(2).sum();
|
||||
preferred.push(usize::from(odd > even));
|
||||
}
|
||||
let mut out = Vec::new();
|
||||
let mut last = None;
|
||||
for flip in [0, 1] {
|
||||
let offsets: Vec<usize> = preferred.iter().map(|&o| o ^ flip).collect();
|
||||
let coarse = pick(region, &offsets)?;
|
||||
match check_coarse(&coarse.region) {
|
||||
Ok(()) => out.push(coarse),
|
||||
Err(e) => last = Some(e),
|
||||
}
|
||||
}
|
||||
if out.is_empty() {
|
||||
return Err(Error::Failed(format!(
|
||||
"the boundary is too coarse for the free quad mesher: the chords joining every other boundary point cross or flip ({}); \
|
||||
refine the boundary divisions",
|
||||
last.map(|e| e.to_string()).unwrap_or_default()
|
||||
)));
|
||||
}
|
||||
Ok(out)
|
||||
}
|
||||
|
||||
/// The half-resolution region, with each coarse point's input id and the input id of the
|
||||
/// boundary point after it (the one the coarse edge skips).
|
||||
struct Coarse {
|
||||
region: Region,
|
||||
to_input: Vec<u32>,
|
||||
skipped: Vec<u32>,
|
||||
/// The input point before each coarse point.
|
||||
before: Vec<u32>,
|
||||
}
|
||||
|
||||
fn pick(region: &Region, offsets: &[usize]) -> Result<Coarse> {
|
||||
let (mut pts, mut p2, mut loops, mut to_input, mut skipped, mut before) =
|
||||
(Vec::new(), Vec::new(), Vec::new(), Vec::new(), Vec::new(), Vec::new());
|
||||
for (l, &off) in region.loops.iter().zip(offsets) {
|
||||
let mut ids = Vec::with_capacity(l.len() / 2);
|
||||
for pos in (off..l.len()).step_by(2) {
|
||||
let (v, after, prior) = match (l.get(pos), l.get((pos + 1) % l.len()), l.get((pos + l.len() - 1) % l.len())) {
|
||||
(Some(&v), Some(&a), Some(&b)) => (v, a, b),
|
||||
_ => return Err(Error::Failed("loop index".into())),
|
||||
};
|
||||
ids.push(pts.len() as u32);
|
||||
pts.push(region.pts.get(v as usize).copied().ok_or_else(|| Error::Failed("point index".into()))?);
|
||||
p2.push(*region.point(v)?);
|
||||
to_input.push(v);
|
||||
skipped.push(after);
|
||||
before.push(prior);
|
||||
}
|
||||
loops.push(ids);
|
||||
}
|
||||
Ok(Coarse { region: Region { pts, p2, loops }, to_input, skipped, before })
|
||||
}
|
||||
|
||||
/// The coarse loops must be simple, keep their orientation and nesting.
|
||||
fn check_coarse(coarse: &Region) -> Result<()> {
|
||||
domain::check_simple(coarse)?;
|
||||
domain::check_orientation(coarse)?;
|
||||
domain::check_nesting(coarse)
|
||||
}
|
||||
|
||||
/// The coarse boundary edges `(a, b, m)`: coarse ids in loop order and the input point `m` the
|
||||
/// edge skips.
|
||||
fn chords(coarse: &Region, skipped: &[u32]) -> Result<Vec<(u32, u32, u32)>> {
|
||||
let mut out = Vec::with_capacity(coarse.pts.len());
|
||||
for (a, b) in domain::segments(&coarse.loops) {
|
||||
out.push((a, b, skipped.get(a as usize).copied().ok_or_else(|| Error::Failed("coarse id".into()))?));
|
||||
}
|
||||
Ok(out)
|
||||
}
|
||||
|
||||
/// How many coarse elements each coarse boundary vertex needs around it. A coarse vertex `v`
|
||||
/// gets one final quadrilateral per incident coarse element, and the corner of the first and last
|
||||
/// of them at `v` opens towards the prescribed neighbours of `v`. A single element would span the
|
||||
/// whole boundary angle at `v`, so a straight or nearly straight (interior angle above 150°) or
|
||||
/// reflex vertex needs two, and one above 300° needs three. Decided with the exact `orient2d` and
|
||||
/// the cosine of the turn.
|
||||
fn fan_needs(region: &Region, to_input: &[u32], before: &[u32], after: &[u32]) -> Result<Vec<usize>> {
|
||||
let mut need = Vec::with_capacity(to_input.len());
|
||||
for ((&v, &a), &b) in to_input.iter().zip(before).zip(after) {
|
||||
let (pa, pv, pb) = (region.point(a)?, region.point(v)?, region.point(b)?);
|
||||
let o = orient2d(pa, pv, pb);
|
||||
let (a, v, b) = (pa.coords(), pv.coords(), pb.coords());
|
||||
let (u, w) = ([v[0] - a[0], v[1] - a[1]], [b[0] - v[0], b[1] - v[1]]);
|
||||
let cos = (u[0] * w[0] + u[1] * w[1]) / ((u[0] * u[0] + u[1] * u[1]).sqrt() * (w[0] * w[0] + w[1] * w[1]).sqrt());
|
||||
// Interior angle 180° − turn: above 150° when the turn is below 30° (cos > √3/2).
|
||||
need.push(match o {
|
||||
Sign::Positive if cos > 0.866_025_403_784_438_6 => 2,
|
||||
Sign::Positive => 1,
|
||||
Sign::Zero => 2,
|
||||
Sign::Negative if cos < -0.5 => 3,
|
||||
Sign::Negative => 2,
|
||||
});
|
||||
}
|
||||
Ok(need)
|
||||
}
|
||||
|
||||
/// The smallest corner scaled Jacobian of a planar quadrilateral (corner `i`:
|
||||
/// `cross(p[i+1] − p[i], p[i−1] − p[i]) / (|…| |…|)`), in `[-1, 1]`.
|
||||
/// A degenerate corner (zero-length edge) scores −1, the worst, so optimizers move away from it.
|
||||
pub(crate) fn min_scaled_jacobian(p: [[f64; 2]; 4]) -> f64 {
|
||||
(0..4).fold(f64::INFINITY, |m, i| m.min(planar_corner(&p, i).1.unwrap_or(-1.0)))
|
||||
}
|
||||
|
||||
/// The smallest corner scaled Jacobian of a recombined quadrilateral (sin 15°): a flatter corner
|
||||
/// would survive the split as a nearly straight angle, which rounding of the mid nodes can turn
|
||||
/// degenerate.
|
||||
const MIN_RECOMBINED_QUALITY: f64 = 0.258_819_045_102_520_74;
|
||||
|
||||
/// Greedy recombination of adjacent triangle pairs into strictly convex quadrilaterals, best
|
||||
/// first by the smallest corner scaled Jacobian (Lee and Lo 1994; Knupp 2001). Returns the
|
||||
/// quadrilaterals (coarse ids, counterclockwise) and the triangles left over.
|
||||
///
|
||||
/// A merge removes one element around the two shared vertices; it is skipped when a boundary
|
||||
/// vertex would drop below the elements it needs ([`fan_needs`]).
|
||||
fn recombine(tri: &Triangulation, need: &[usize]) -> Result<(Vec<[u32; 4]>, Vec<[u32; 3]>)> {
|
||||
let mut count = vec![0usize; tri.pts.len()];
|
||||
for t in &tri.tris {
|
||||
for &v in t {
|
||||
if let Some(c) = count.get_mut(v as usize) {
|
||||
*c += 1;
|
||||
}
|
||||
}
|
||||
}
|
||||
let mut edges: Vec<(u32, u32, u32, usize)> = Vec::with_capacity(3 * tri.tris.len());
|
||||
for (t, v) in tri.tris.iter().enumerate() {
|
||||
for i in 0..3 {
|
||||
let (a, b) = (v[(i + 1) % 3], v[(i + 2) % 3]);
|
||||
edges.push((a.min(b), a.max(b), t as u32, i));
|
||||
}
|
||||
}
|
||||
edges.sort_unstable();
|
||||
let pt = |v: u32| tri.p2.get(v as usize).ok_or_else(|| Error::Failed("point index".into()));
|
||||
let xy = |v: u32| tri.pts.get(v as usize).copied().ok_or_else(|| Error::Failed("point index".into()));
|
||||
let mut candidates: Vec<(f64, u32, u32, [u32; 4], (u32, u32))> = Vec::new();
|
||||
for w in edges.windows(2) {
|
||||
let [(a0, b0, t1, i), (a1, b1, t2, j)] = w else { continue };
|
||||
if (a0, b0) != (a1, b1) {
|
||||
continue;
|
||||
}
|
||||
let (v1, v2) = match (tri.tris.get(*t1 as usize), tri.tris.get(*t2 as usize)) {
|
||||
(Some(x), Some(y)) => (x, y),
|
||||
_ => continue,
|
||||
};
|
||||
let (p, q, r, s) = (v1[*i], v1[(i + 1) % 3], v1[(i + 2) % 3], v2[*j]);
|
||||
let quad = [p, q, s, r];
|
||||
if !turns_left([pt(p)?, pt(q)?, pt(s)?, pt(r)?]) {
|
||||
continue;
|
||||
}
|
||||
let quality = min_scaled_jacobian([xy(p)?, xy(q)?, xy(s)?, xy(r)?]);
|
||||
if quality < MIN_RECOMBINED_QUALITY {
|
||||
continue;
|
||||
}
|
||||
candidates.push((quality, *t1, *t2, quad, (q, r)));
|
||||
}
|
||||
candidates.sort_by(|x, y| y.0.total_cmp(&x.0).then((x.1, x.2).cmp(&(y.1, y.2))));
|
||||
let mut used = vec![false; tri.tris.len()];
|
||||
let mut quads = Vec::new();
|
||||
for (_, t1, t2, quad, (q, r)) in candidates {
|
||||
let (Some(&u1), Some(&u2)) = (used.get(t1 as usize), used.get(t2 as usize)) else { continue };
|
||||
if u1 || u2 {
|
||||
continue;
|
||||
}
|
||||
let keeps = |v: u32| count.get(v as usize).copied().unwrap_or(0) > need.get(v as usize).copied().unwrap_or(0);
|
||||
if !keeps(q) || !keeps(r) {
|
||||
continue;
|
||||
}
|
||||
for v in [q, r] {
|
||||
if let Some(c) = count.get_mut(v as usize) {
|
||||
*c -= 1;
|
||||
}
|
||||
}
|
||||
for t in [t1, t2] {
|
||||
if let Some(u) = used.get_mut(t as usize) {
|
||||
*u = true;
|
||||
}
|
||||
}
|
||||
quads.push(quad);
|
||||
}
|
||||
let leftovers = tri.tris.iter().zip(&used).filter(|(_, u)| !**u).map(|(t, _)| *t).collect();
|
||||
Ok((quads, leftovers))
|
||||
}
|
||||
|
||||
/// The uniform split: every coarse edge gets a mid node (on the boundary, the prescribed point
|
||||
/// it skipped), every element a centre node; a quadrilateral becomes four quadrilaterals, a
|
||||
/// triangle three. Node order: the input boundary points (input order), the interior coarse
|
||||
/// vertices, the interior edge mid nodes, the centres.
|
||||
fn split(
|
||||
region: &Region,
|
||||
tri: &Triangulation,
|
||||
to_input: &[u32],
|
||||
boundary_mid: &BTreeMap<(u32, u32), u32>,
|
||||
quads: &[[u32; 4]],
|
||||
leftovers: &[[u32; 3]],
|
||||
) -> Result<(Vec<[f64; 2]>, Vec<[u32; 4]>)> {
|
||||
let mut nodes = region.pts.clone();
|
||||
let mut final_id = vec![u32::MAX; tri.pts.len()];
|
||||
for (c, &input) in to_input.iter().enumerate() {
|
||||
if let Some(f) = final_id.get_mut(c) {
|
||||
*f = input;
|
||||
}
|
||||
}
|
||||
let xy = |v: u32| tri.pts.get(v as usize).copied().ok_or_else(|| Error::Failed("point index".into()));
|
||||
let elements: Vec<&[u32]> = quads.iter().map(|q| q.as_slice()).chain(leftovers.iter().map(|t| t.as_slice())).collect();
|
||||
for e in &elements {
|
||||
for &v in e.iter() {
|
||||
let f = final_id.get_mut(v as usize).ok_or_else(|| Error::Failed("vertex id".into()))?;
|
||||
if *f == u32::MAX {
|
||||
*f = nodes.len() as u32;
|
||||
nodes.push(xy(v)?);
|
||||
}
|
||||
}
|
||||
}
|
||||
let fid = |v: u32| final_id.get(v as usize).copied().filter(|&f| f != u32::MAX).ok_or_else(|| Error::Failed("vertex id".into()));
|
||||
let mut mids: BTreeMap<(u32, u32), u32> = BTreeMap::new();
|
||||
for e in &elements {
|
||||
for (i, &a) in e.iter().enumerate() {
|
||||
let b = e.get((i + 1) % e.len()).copied().ok_or_else(|| Error::Failed("element".into()))?;
|
||||
let key = (a.min(b), a.max(b));
|
||||
if let std::collections::btree_map::Entry::Vacant(slot) = mids.entry(key) {
|
||||
let id = match boundary_mid.get(&key) {
|
||||
Some(&m) => m,
|
||||
None => {
|
||||
let (pa, pb) = (xy(a)?, xy(b)?);
|
||||
nodes.push([0.5 * (pa[0] + pb[0]), 0.5 * (pa[1] + pb[1])]);
|
||||
nodes.len() as u32 - 1
|
||||
}
|
||||
};
|
||||
slot.insert(id);
|
||||
}
|
||||
}
|
||||
}
|
||||
let mid = |a: u32, b: u32| mids.get(&(a.min(b), a.max(b))).copied().ok_or_else(|| Error::Failed("mid node".into()));
|
||||
let mut out = Vec::with_capacity(4 * quads.len() + 3 * leftovers.len());
|
||||
for e in &elements {
|
||||
let k = e.len();
|
||||
let mut c = [0.0, 0.0];
|
||||
for &v in e.iter() {
|
||||
let p = xy(v)?;
|
||||
c = [c[0] + p[0], c[1] + p[1]];
|
||||
}
|
||||
nodes.push([c[0] / k as f64, c[1] / k as f64]);
|
||||
let centre = nodes.len() as u32 - 1;
|
||||
for (i, &v) in e.iter().enumerate() {
|
||||
let next = e.get((i + 1) % k).copied().ok_or_else(|| Error::Failed("element".into()))?;
|
||||
let prev = e.get((i + k - 1) % k).copied().ok_or_else(|| Error::Failed("element".into()))?;
|
||||
out.push([fid(v)?, mid(v, next)?, centre, mid(prev, v)?]);
|
||||
}
|
||||
}
|
||||
Ok((nodes, out))
|
||||
}
|
||||
@@ -0,0 +1,121 @@
|
||||
//! Where the half-resolution triangulation may place Steiner points.
|
||||
//!
|
||||
//! The half-resolution region replaces each pair of boundary edges `a–m–b` by the chord `a–b`;
|
||||
//! the split later puts `m` back as the chord's mid node. The final mesh therefore covers the true
|
||||
//! region, not the coarse one, and a Steiner point must respect the true boundary: it may not lie
|
||||
//! in a *cap* (the triangle `a, b, m` when `m` lies inside the coarse region, which the true region
|
||||
//! excludes), and it keeps a clearance from the true edges `a–m`, `m–b` of at least half the chord
|
||||
//! (so the elements along the boundary keep the boundary's spacing and can absorb `m`'s offset
|
||||
//! from the chord). Lookups go through a uniform grid; the cap test is exact (`orient2d`).
|
||||
|
||||
use crate::predicates::{Point2, Sign, orient2d};
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// Above this many grid cells a clearance query scans every segment instead.
|
||||
const MAX_QUERY_CELLS: usize = 1024;
|
||||
|
||||
pub(crate) struct Guard {
|
||||
lo: [f64; 2],
|
||||
step: [f64; 2],
|
||||
g: usize,
|
||||
/// True boundary edges with their own clearance radius.
|
||||
segs: Vec<([f64; 2], [f64; 2], f64)>,
|
||||
/// Counterclockwise caps `(a, b, m)`.
|
||||
caps: Vec<[Point2; 3]>,
|
||||
seg_cells: Vec<Vec<u32>>,
|
||||
cap_cells: Vec<Vec<u32>>,
|
||||
}
|
||||
|
||||
fn distance(a: [f64; 2], b: [f64; 2], p: [f64; 2]) -> f64 {
|
||||
let (d, w) = ([b[0] - a[0], b[1] - a[1]], [p[0] - a[0], p[1] - a[1]]);
|
||||
let len2 = d[0] * d[0] + d[1] * d[1];
|
||||
let t = if len2 > 0.0 { ((w[0] * d[0] + w[1] * d[1]) / len2).clamp(0.0, 1.0) } else { 0.0 };
|
||||
((w[0] - t * d[0]).powi(2) + (w[1] - t * d[1]).powi(2)).sqrt()
|
||||
}
|
||||
|
||||
impl Guard {
|
||||
/// The guard of chords `(a, b, m)`: coarse boundary edge `a → b` (region on its left) and the
|
||||
/// skipped boundary point `m`.
|
||||
pub fn new(chords: &[(Point2, Point2, Point2)]) -> Result<Guard> {
|
||||
let (lo, hi) = super::domain::bbox(chords.iter().flat_map(|(a, b, m)| [a.coords(), b.coords(), m.coords()]));
|
||||
if chords.is_empty() {
|
||||
return Err(Error::Failed("guard without a boundary".into()));
|
||||
}
|
||||
let g = ((2.0 * chords.len() as f64).sqrt().ceil() as usize).clamp(1, 1024);
|
||||
let step = [((hi[0] - lo[0]) / g as f64).max(f64::MIN_POSITIVE), ((hi[1] - lo[1]) / g as f64).max(f64::MIN_POSITIVE)];
|
||||
let mut guard =
|
||||
Guard { lo, step, g, segs: Vec::new(), caps: Vec::new(), seg_cells: vec![Vec::new(); g * g], cap_cells: vec![Vec::new(); g * g] };
|
||||
for (a, b, m) in chords {
|
||||
let (pa, pb, pm) = (a.coords(), b.coords(), m.coords());
|
||||
let radius = 0.5 * ((pb[0] - pa[0]).powi(2) + (pb[1] - pa[1]).powi(2)).sqrt();
|
||||
for (s, t) in [(pa, pm), (pm, pb)] {
|
||||
let id = guard.segs.len() as u32;
|
||||
guard.segs.push((s, t, radius));
|
||||
let (x0, y0) = guard.cell([s[0].min(t[0]) - radius, s[1].min(t[1]) - radius]);
|
||||
let (x1, y1) = guard.cell([s[0].max(t[0]) + radius, s[1].max(t[1]) + radius]);
|
||||
guard.register(true, id, (x0, y0), (x1, y1));
|
||||
}
|
||||
if orient2d(a, b, m) == Sign::Positive {
|
||||
let id = guard.caps.len() as u32;
|
||||
guard.caps.push([*a, *b, *m]);
|
||||
let (x0, y0) = guard.cell([pa[0].min(pb[0]).min(pm[0]), pa[1].min(pb[1]).min(pm[1])]);
|
||||
let (x1, y1) = guard.cell([pa[0].max(pb[0]).max(pm[0]), pa[1].max(pb[1]).max(pm[1])]);
|
||||
guard.register(false, id, (x0, y0), (x1, y1));
|
||||
}
|
||||
}
|
||||
Ok(guard)
|
||||
}
|
||||
|
||||
fn cell(&self, p: [f64; 2]) -> (usize, usize) {
|
||||
let f = |x: f64, lo: f64, step: f64| {
|
||||
let c = ((x - lo) / step).floor();
|
||||
if c.is_nan() || c < 0.0 { 0 } else { (c as usize).min(self.g - 1) }
|
||||
};
|
||||
(f(p[0], self.lo[0], self.step[0]), f(p[1], self.lo[1], self.step[1]))
|
||||
}
|
||||
|
||||
fn register(&mut self, seg: bool, id: u32, from: (usize, usize), to: (usize, usize)) {
|
||||
let cells = if seg { &mut self.seg_cells } else { &mut self.cap_cells };
|
||||
for y in from.1..=to.1 {
|
||||
for x in from.0..=to.0 {
|
||||
if let Some(c) = cells.get_mut(y * self.g + x) {
|
||||
c.push(id);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/// Whether a Steiner point may go at `p`: outside every cap, and farther than both the
|
||||
/// segment's own radius and `clearance` from every true boundary edge.
|
||||
pub fn allows(&self, p: &Point2, clearance: f64) -> bool {
|
||||
let c = p.coords();
|
||||
let (cx, cy) = self.cell(c);
|
||||
if let Some(caps) = self.cap_cells.get(cy * self.g + cx) {
|
||||
for &k in caps {
|
||||
if let Some([a, b, m]) = self.caps.get(k as usize)
|
||||
&& orient2d(a, b, p) != Sign::Negative
|
||||
&& orient2d(b, m, p) != Sign::Negative
|
||||
&& orient2d(m, a, p) != Sign::Negative
|
||||
{
|
||||
return false;
|
||||
}
|
||||
}
|
||||
}
|
||||
let too_close = |&(a, b, r): &([f64; 2], [f64; 2], f64)| distance(a, b, c) < r.max(clearance);
|
||||
let (x0, y0) = self.cell([c[0] - clearance, c[1] - clearance]);
|
||||
let (x1, y1) = self.cell([c[0] + clearance, c[1] + clearance]);
|
||||
if (x1 - x0 + 1) * (y1 - y0 + 1) > MAX_QUERY_CELLS {
|
||||
return !self.segs.iter().any(too_close);
|
||||
}
|
||||
for y in y0..=y1 {
|
||||
for x in x0..=x1 {
|
||||
for &k in self.seg_cells.get(y * self.g + x).map(Vec::as_slice).unwrap_or(&[]) {
|
||||
if self.segs.get(k as usize).is_some_and(too_close) {
|
||||
return false;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
true
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,287 @@
|
||||
//! Planar all-quadrilateral meshing: mapped (transfinite) meshes of four-sided regions and free
|
||||
//! meshes of polygons with holes, on prescribed boundary nodes.
|
||||
//!
|
||||
//! These are the face meshers under the mapped and swept hexahedral meshers and shell meshing.
|
||||
//! Both take the boundary nodes as given (shared edges must share nodes), put them first in the
|
||||
//! output in input order, and return counterclockwise `Quad4` connectivity whose every element is
|
||||
//! proven valid by the exact corner test ([`crate::quality::planar_quad_validity`]).
|
||||
//!
|
||||
//! # Mapped: [`mapped_quad_mesh`], [`transfinite_grid`]
|
||||
//! Transfinite interpolation (Gordon and Hall 1973), see [`tfi`]. Opposite sides need equal
|
||||
//! division counts; a region whose TFI folds (strongly concave sides) is an error, not a tangled
|
||||
//! mesh.
|
||||
//!
|
||||
//! # Free: [`free_quad_mesh`]
|
||||
//! **Method.** An indirect tri-to-quad method in three steps:
|
||||
//! 1. Keep every other boundary point of each loop (the parity class with the sharper turns, so
|
||||
//! corners at even spacing survive) and build the constrained Delaunay triangulation of that
|
||||
//! half-resolution region, refined to twice the target size ([`cdt`]: Bowyer–Watson, Sloan
|
||||
//! 1993 recovery, Ruppert 1995 refinement without segment splitting).
|
||||
//! 2. Recombine adjacent triangles into strictly convex quadrilaterals, greedily, best first by
|
||||
//! the smallest corner scaled Jacobian (Lee and Lo, *A new scheme for the generation of a
|
||||
//! graded quadrilateral mesh*, Computers & Structures 52(5):847–857, 1994).
|
||||
//! 3. Split every element once: a quadrilateral into four, a leftover triangle into three, through
|
||||
//! edge mid nodes and a centre node. On a boundary edge the mid node is the prescribed point the
|
||||
//! half-resolution loop skipped, so the final boundary nodes are exactly the input points. The
|
||||
//! split is the Catmull–Clark topology (Catmull and Clark, CAD 10(6):350–355, 1978) and appears
|
||||
//! as the all-quad completion of recombination in Lee and Lo 1994.
|
||||
//!
|
||||
//! Then constrained Laplacian smoothing ([`smooth`]) and the exact validity gate.
|
||||
//!
|
||||
//! **Why this method.** The alternatives considered, from their papers:
|
||||
//! - *Q-Morph* (Owen, Staten, Canann, Saigal, *Q-Morph: an indirect approach to advancing front
|
||||
//! quad meshing*, IJNME 44(9):1317–1340, 1999) gives better-aligned rows along the boundary, but
|
||||
//! needs front-edge state classification, side-edge recovery by swaps, seam closing and special
|
||||
//! handling where fronts collide; the paper reports the cases that need it, and each is a place
|
||||
//! to fail on concave faces with many holes. An odd front remainder still ends in a triangle
|
||||
//! to be removed.
|
||||
//! - *Paving* (Blacker and Stephenson, IJNME 32(4):811–847, 1991) has the same front-collision
|
||||
//! problem with no background triangulation to fall back on; it is the highest-risk choice.
|
||||
//! - *Blossom-Quad* (Remacle, Lambrechts, Seny, Marchandise, Johnen, Geuzaine, *Blossom-Quad: a
|
||||
//! non-uniform quadrilateral mesh generator using a minimum-cost perfect-matching algorithm*,
|
||||
//! IJNME 89(9):1102–1119, 2012; paper only, Gmsh's code is GPL) gives the best matching, but a
|
||||
//! perfect matching of the triangle graph need not exist, so it still needs a fallback, and it
|
||||
//! needs Edmonds' weighted blossom algorithm (cubic in the worst case).
|
||||
//!
|
||||
//! The split method is **all-quad by construction**: any triangulation of the half-resolution
|
||||
//! region becomes quadrilaterals only, whatever the matching found, so the only requirement is an
|
||||
//! even number of boundary points on every loop. Robustness on concave regions with holes is
|
||||
//! that of the constrained Delaunay triangulation, which exists for every valid input and is
|
||||
//! built here with exact predicates. The implementation risk is low: one triangulator (which the
|
||||
//! surface and tetrahedral work need anyway), a greedy matching and a local split. The cost is
|
||||
//! quality: the split creates irregular nodes (valence 3 at triangle centres, 6 at some coarse
|
||||
//! vertices) and triangle-born quads have a 120° corner; smoothing removes most of the
|
||||
//! distortion. If that proves too weak, the greedy matching can be replaced by Blossom-Quad's
|
||||
//! minimum-cost matching without touching the rest (fewer leftover triangles, same guarantee).
|
||||
//!
|
||||
//! **Requirements and failure modes.**
|
||||
//! - Every loop has an even number of points, at least 6 when there is more than one loop (a
|
||||
//! single 4-point loop is meshed as one quadrilateral). An odd loop is an error, never repaired:
|
||||
//! the points are prescribed by neighbouring faces, so adding one would break conformity. Choose
|
||||
//! divisions with [`crate::size::loop_divisions`] or [`crate::size::even_divisions`].
|
||||
//! - The chords joining every other boundary point must not cross (a boundary too coarse for a
|
||||
//! thin feature); both parity classes are tried before this is reported.
|
||||
//! - Every output element is checked with exact predicates; a mesh that smoothing cannot make
|
||||
//! valid is an error.
|
||||
//! - Hostile input (NaN, coordinates beyond the model limit, repeated points, zero-length edges,
|
||||
//! crossing or touching loops, holes outside the outer loop or inside another hole) is an
|
||||
//! [`Error::Invalid`]; size and work limits are [`Error::Limit`].
|
||||
|
||||
mod cdt;
|
||||
mod domain;
|
||||
mod free;
|
||||
mod guard;
|
||||
mod smooth;
|
||||
pub mod tfi;
|
||||
|
||||
pub use domain::{MAX_BOUNDARY_POINTS, MAX_LOOPS};
|
||||
pub use tfi::{MAX_GRID_NODES, transfinite_grid};
|
||||
|
||||
use crate::predicates::{Point2, Sign, snap_to_domain};
|
||||
use crate::quality::{planar_corner, quad_axes, turns_left};
|
||||
use crate::size::SizeField;
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// Limits and smoothing of the quad meshers.
|
||||
#[derive(Debug, Clone, PartialEq)]
|
||||
pub struct QuadOptions {
|
||||
/// The most quadrilaterals of one mesh ([`Error::Limit`] beyond).
|
||||
pub max_elements: usize,
|
||||
/// Smoothing sweeps of the free mesher (0 turns smoothing off).
|
||||
pub smoothing_sweeps: usize,
|
||||
}
|
||||
|
||||
impl Default for QuadOptions {
|
||||
fn default() -> Self {
|
||||
QuadOptions { max_elements: 4_000_000, smoothing_sweeps: 10 }
|
||||
}
|
||||
}
|
||||
|
||||
/// What a quad mesher produced and how good it is.
|
||||
#[derive(Debug, Clone, PartialEq)]
|
||||
pub struct QuadStats {
|
||||
/// Smallest corner scaled Jacobian over the mesh (1 for squares).
|
||||
pub min_scaled_jacobian: f64,
|
||||
/// Largest skew (0 for rectangles).
|
||||
pub max_skew: f64,
|
||||
/// Largest aspect ratio (1 for squares).
|
||||
pub max_aspect_ratio: f64,
|
||||
/// Smoothing sweeps run, and whether smoothing converged before its sweep limit.
|
||||
pub smoothing_sweeps: usize,
|
||||
pub smoothing_converged: bool,
|
||||
/// Free mesher: triangle pairs recombined, and triangles split on their own.
|
||||
pub recombined: usize,
|
||||
pub leftover_triangles: usize,
|
||||
}
|
||||
|
||||
impl Default for QuadStats {
|
||||
fn default() -> Self {
|
||||
QuadStats {
|
||||
min_scaled_jacobian: 1.0,
|
||||
max_skew: 0.0,
|
||||
max_aspect_ratio: 1.0,
|
||||
smoothing_sweeps: 0,
|
||||
smoothing_converged: true,
|
||||
recombined: 0,
|
||||
leftover_triangles: 0,
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/// A planar all-quadrilateral mesh.
|
||||
#[derive(Debug, Clone, PartialEq)]
|
||||
pub struct QuadMesh {
|
||||
/// Node coordinates. `nodes[..boundary_nodes]` are the input boundary points, unchanged and
|
||||
/// in input order (loop by loop), so node `k` of the mesh is boundary point `k`.
|
||||
pub nodes: Vec<[f64; 2]>,
|
||||
/// Counterclockwise corner node ids (the `Quad4` node order).
|
||||
pub quads: Vec<[u32; 4]>,
|
||||
pub boundary_nodes: usize,
|
||||
pub stats: QuadStats,
|
||||
}
|
||||
|
||||
/// A mapped mesh with its structure: node `(i, j)`, `0 ≤ i ≤ nu`, `0 ≤ j ≤ nv`, is
|
||||
/// `mesh.nodes[grid[j·(nu + 1) + i]]`, with `i` running from corner 0 to corner 1 and `j` from
|
||||
/// corner 1 to corner 2 (for a sweep: the source face's structured layout).
|
||||
#[derive(Debug, Clone, PartialEq)]
|
||||
pub struct MappedMesh {
|
||||
pub mesh: QuadMesh,
|
||||
pub nu: usize,
|
||||
pub nv: usize,
|
||||
pub grid: Vec<u32>,
|
||||
}
|
||||
|
||||
/// Free all-quadrilateral mesh of a planar region (see the module docs for the method).
|
||||
///
|
||||
/// `loops[0]` is the outer boundary, the others are holes; each loop lists its boundary nodes
|
||||
/// once (not closed), in either orientation, with an even count. `size` sets the interior
|
||||
/// density; the boundary density is the given nodes.
|
||||
pub fn free_quad_mesh(loops: &[Vec<[f64; 2]>], size: &dyn SizeField, opts: &QuadOptions) -> Result<QuadMesh> {
|
||||
free::mesh(loops, size, opts)
|
||||
}
|
||||
|
||||
/// Mapped mesh of a four-sided region: `boundary` lists the region's boundary nodes once (either
|
||||
/// orientation), and `corners` are the positions in it of the four corners in boundary order.
|
||||
/// Opposite sides need the same number of divisions. The grid is transfinite interpolation
|
||||
/// ([`transfinite_grid`]); a grid that folds is an error.
|
||||
pub fn mapped_quad_mesh(boundary: &[[f64; 2]], corners: [usize; 4], opts: &QuadOptions) -> Result<MappedMesh> {
|
||||
let region = domain::region(&[boundary.to_vec()])?;
|
||||
let n = boundary.len();
|
||||
if let Some(c) = corners.iter().find(|&&c| c >= n) {
|
||||
return Err(Error::Invalid(format!("corner position {c} is outside the boundary of {n} points")));
|
||||
}
|
||||
let c0 = corners[0];
|
||||
let d = corners.map(|c| (c + n - c0) % n);
|
||||
if !(d[0] == 0 && d[0] < d[1] && d[1] < d[2] && d[2] < d[3]) {
|
||||
return Err(Error::Invalid("the corners must be four distinct positions in boundary order".into()));
|
||||
}
|
||||
let (nu, nv) = (d[1], d[2] - d[1]);
|
||||
if d[3] - d[2] != nu || n - d[3] != nv {
|
||||
return Err(Error::Invalid(format!("opposite sides need the same number of divisions: {} and {}, {} and {}", nu, d[3] - d[2], nv, n - d[3])));
|
||||
}
|
||||
let at = |k: usize| region.pts.get((c0 + k) % n).copied().ok_or_else(|| Error::Failed("boundary index".into()));
|
||||
let side = |from: usize, len: usize, forward: bool| -> Result<Vec<[f64; 2]>> {
|
||||
(0..=len).map(|k| if forward { at(from + k) } else { at(from + n - k) }).collect()
|
||||
};
|
||||
// Boundary positions relative to corner 0: bottom 0..nu, right nu..nu+nv, top runs backwards
|
||||
// from d[3], left backwards from 0 (= n).
|
||||
let bottom = side(0, nu, true)?;
|
||||
let right = side(d[1], nv, true)?;
|
||||
let top = side(d[3], nu, false)?;
|
||||
let left = side(n, nv, false)?;
|
||||
let points = transfinite_grid(&bottom, &right, &top, &left)?;
|
||||
// Node ids: boundary nodes keep their input positions; interior nodes follow, row by row.
|
||||
let mut nodes = region.pts.clone();
|
||||
let mut grid = Vec::with_capacity(points.len());
|
||||
for j in 0..=nv {
|
||||
for i in 0..=nu {
|
||||
let pos = if j == 0 {
|
||||
Some(i)
|
||||
} else if i == nu {
|
||||
Some(nu + j)
|
||||
} else if j == nv {
|
||||
Some(d[3] - i)
|
||||
} else if i == 0 {
|
||||
Some((n - j) % n)
|
||||
} else {
|
||||
None
|
||||
};
|
||||
let id = match pos {
|
||||
Some(p) => ((c0 + p) % n) as u32,
|
||||
None => {
|
||||
let x = points.get(j * (nu + 1) + i).copied().ok_or_else(|| Error::Failed("grid index".into()))?;
|
||||
nodes.push(domain::admissible(x)?);
|
||||
nodes.len() as u32 - 1
|
||||
}
|
||||
};
|
||||
grid.push(id);
|
||||
}
|
||||
}
|
||||
let ccw = domain::loop_orientation(®ion, &(0..n as u32).collect::<Vec<_>>())? == Sign::Positive;
|
||||
let g = |i: usize, j: usize| grid.get(j * (nu + 1) + i).copied().ok_or_else(|| Error::Failed("grid index".into()));
|
||||
let mut quads = Vec::with_capacity(nu * nv);
|
||||
for j in 0..nv {
|
||||
for i in 0..nu {
|
||||
let (a, b, c, e) = (g(i, j)?, g(i + 1, j)?, g(i + 1, j + 1)?, g(i, j + 1)?);
|
||||
quads.push(if ccw { [a, b, c, e] } else { [a, e, c, b] });
|
||||
}
|
||||
}
|
||||
if quads.len() > opts.max_elements {
|
||||
return Err(Error::Limit(format!("the mesh would have {} quadrilaterals; the limit is {}", quads.len(), opts.max_elements)));
|
||||
}
|
||||
let mesh = finish(nodes, quads, n, &QuadOptions { smoothing_sweeps: 0, ..opts.clone() }, QuadStats::default())?;
|
||||
Ok(MappedMesh { mesh, nu, nv, grid })
|
||||
}
|
||||
|
||||
/// Smooth (free nodes only), gate every element with the exact validity test and measure.
|
||||
pub(crate) fn finish(
|
||||
mut nodes: Vec<[f64; 2]>,
|
||||
quads: Vec<[u32; 4]>,
|
||||
boundary_nodes: usize,
|
||||
opts: &QuadOptions,
|
||||
mut stats: QuadStats,
|
||||
) -> Result<QuadMesh> {
|
||||
// Constructed nodes (mid nodes, centres) stay in the predicates' domain.
|
||||
for n in nodes.iter_mut() {
|
||||
*n = [snap_to_domain(n[0]), snap_to_domain(n[1])];
|
||||
}
|
||||
if opts.smoothing_sweeps > 0 {
|
||||
let (sweeps, converged) = smooth::smooth(&mut nodes, &quads, boundary_nodes, opts.smoothing_sweeps)?;
|
||||
stats.smoothing_sweeps = sweeps;
|
||||
stats.smoothing_converged = converged;
|
||||
}
|
||||
let mut invalid = 0usize;
|
||||
let mut first_invalid = None;
|
||||
let (mut sj, mut skew, mut aspect) = (f64::INFINITY, 0.0_f64, 1.0_f64);
|
||||
for (e, q) in quads.iter().enumerate() {
|
||||
let mut c = [[0.0; 2]; 4];
|
||||
for (k, &v) in q.iter().enumerate() {
|
||||
c[k] = nodes.get(v as usize).copied().ok_or_else(|| Error::Failed("node index".into()))?;
|
||||
}
|
||||
let [a, b, d, f] = c.map(|x| Point2::new(x[0], x[1]));
|
||||
if !turns_left([&a?, &b?, &d?, &f?]) {
|
||||
invalid += 1;
|
||||
first_invalid.get_or_insert(e);
|
||||
}
|
||||
sj = sj.min((0..4).fold(f64::INFINITY, |m, i| m.min(planar_corner(&c, i).1.unwrap_or(0.0))));
|
||||
let (r, k) = quad_axes(c.map(|x| [x[0], x[1], 0.0]));
|
||||
aspect = aspect.max(r);
|
||||
skew = skew.max(k);
|
||||
}
|
||||
if invalid > 0 {
|
||||
let at = first_invalid.and_then(|e| quads.get(e)).and_then(|q| nodes.get(q[0] as usize)).copied().unwrap_or_default();
|
||||
return Err(Error::Failed(format!(
|
||||
"{invalid} of {} quadrilaterals are inverted or degenerate (first near {at:?}); refine the boundary there or use a finer size",
|
||||
quads.len()
|
||||
)));
|
||||
}
|
||||
if !quads.is_empty() {
|
||||
stats.min_scaled_jacobian = sj;
|
||||
stats.max_skew = skew;
|
||||
stats.max_aspect_ratio = aspect;
|
||||
}
|
||||
Ok(QuadMesh { nodes, quads, boundary_nodes, stats })
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests;
|
||||
@@ -0,0 +1,250 @@
|
||||
//! Constrained Laplacian smoothing of a planar quadrilateral mesh.
|
||||
//!
|
||||
//! Each free node moves towards the average of its edge neighbours (Laplacian smoothing), and
|
||||
//! the move is kept only when it does not lower the smallest scaled Jacobian of the node's
|
||||
//! quadrilaterals and keeps them valid by the exact corner test ("smart" Laplacian: Freitag,
|
||||
//! *On combining Laplacian and optimization-based mesh smoothing techniques*, AMD-Vol. 220,
|
||||
//! ASME 1997; Canann, Tristano, Staten, *An approach to combined Laplacian and optimization-based
|
||||
//! smoothing for triangular, quadrilateral, and quad-dominant meshes*, IMR 1998). Half and quarter
|
||||
//! steps are tried when the full step is refused. Nodes of quadrilaterals that are already
|
||||
//! invalid accept any step that raises their smallest corner area (see below).
|
||||
//!
|
||||
//! Laplacian smoothing cannot untangle a node whose average position is itself bad, and the
|
||||
//! nodes next to a sharp prescribed boundary point often are. Every node whose quadrilaterals
|
||||
//! remain inverted or poor (smallest scaled Jacobian below [`POOR`]) then gets optimization-based
|
||||
//! smoothing (Freitag 1997) by a compass search (eight directions, step halved on failure,
|
||||
//! bounded). An inverted patch is untangled first by maximizing its smallest corner area, which is
|
||||
//! linear in the node's position, so its maximum lies in the feasible region when there is one
|
||||
//! (Freitag and Plassmann, *Local optimization-based simplicial mesh untangling and
|
||||
//! improvement*, IJNME 49:109–125, 2000); a valid patch then maximizes its smallest scaled
|
||||
//! Jacobian without leaving the valid set.
|
||||
//!
|
||||
//! Gauss–Seidel order by node id; bounded sweeps; deterministic.
|
||||
|
||||
use crate::predicates::{Point2, snap_to_domain};
|
||||
use crate::quality::{planar_corner, turns_left};
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// Compressed adjacency: `items[start[v]..start[v + 1]]`.
|
||||
struct Csr {
|
||||
start: Vec<usize>,
|
||||
items: Vec<u32>,
|
||||
}
|
||||
|
||||
impl Csr {
|
||||
fn of(n: usize, pairs: &mut [(u32, u32)]) -> Csr {
|
||||
pairs.sort_unstable();
|
||||
let mut start = vec![0usize; n + 1];
|
||||
let mut items = Vec::with_capacity(pairs.len());
|
||||
let mut prev = None;
|
||||
for &(a, b) in pairs.iter() {
|
||||
if prev == Some((a, b)) {
|
||||
continue;
|
||||
}
|
||||
prev = Some((a, b));
|
||||
if let Some(s) = start.get_mut(a as usize + 1) {
|
||||
*s += 1;
|
||||
}
|
||||
items.push(b);
|
||||
}
|
||||
for i in 0..n {
|
||||
let before = start.get(i).copied().unwrap_or(0);
|
||||
if let Some(s) = start.get_mut(i + 1) {
|
||||
*s += before;
|
||||
}
|
||||
}
|
||||
Csr { start, items }
|
||||
}
|
||||
|
||||
fn get(&self, v: usize) -> &[u32] {
|
||||
match (self.start.get(v), self.start.get(v + 1)) {
|
||||
(Some(&a), Some(&b)) => self.items.get(a..b).unwrap_or(&[]),
|
||||
_ => &[],
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn corners(nodes: &[[f64; 2]], q: &[u32; 4], v: u32, at: [f64; 2]) -> Result<[[f64; 2]; 4]> {
|
||||
let mut c = [[0.0; 2]; 4];
|
||||
for (k, &u) in q.iter().enumerate() {
|
||||
c[k] = if u == v { at } else { nodes.get(u as usize).copied().ok_or_else(|| Error::Failed("node index".into()))? };
|
||||
}
|
||||
Ok(c)
|
||||
}
|
||||
|
||||
/// The state of `v`'s quadrilaterals with `v` at some position.
|
||||
#[derive(Debug, Clone, Copy)]
|
||||
struct Local {
|
||||
/// Smallest scaled Jacobian.
|
||||
sj: f64,
|
||||
/// Smallest unnormalized corner cross product among the corners that move with `v` (linear
|
||||
/// in `v`'s position: the untangling objective).
|
||||
area: f64,
|
||||
/// All valid by the exact test.
|
||||
valid: bool,
|
||||
}
|
||||
|
||||
impl Local {
|
||||
/// Valid beats invalid; then a valid patch ranks by its worst scaled Jacobian and an invalid
|
||||
/// one by its worst corner area.
|
||||
fn key(&self) -> (bool, f64) {
|
||||
(self.valid, if self.valid { self.sj } else { self.area })
|
||||
}
|
||||
fn better(&self, than: &Local) -> bool {
|
||||
let (a, b) = (self.key(), than.key());
|
||||
(a.0 && !b.0) || (a.0 == b.0 && a.1 > b.1)
|
||||
}
|
||||
fn no_worse(&self, than: &Local) -> bool {
|
||||
!than.better(self)
|
||||
}
|
||||
}
|
||||
|
||||
/// `v`'s quadrilaterals with `v` at `at`.
|
||||
fn local(nodes: &[[f64; 2]], quads: &[[u32; 4]], incident: &[u32], v: u32, at: [f64; 2]) -> Result<Local> {
|
||||
let mut l = Local { sj: f64::INFINITY, area: f64::INFINITY, valid: true };
|
||||
for &e in incident {
|
||||
let quad = quads.get(e as usize).ok_or_else(|| Error::Failed("quad index".into()))?;
|
||||
let c = corners(nodes, quad, v, at)?;
|
||||
let k = quad.iter().position(|&u| u == v).unwrap_or(0);
|
||||
for i in 0..4 {
|
||||
let (cross, scaled) = planar_corner(&c, i);
|
||||
l.sj = l.sj.min(scaled.unwrap_or(-1.0));
|
||||
// The corner opposite `v` does not move with it.
|
||||
if i != (k + 2) % 4 {
|
||||
l.area = l.area.min(cross);
|
||||
}
|
||||
}
|
||||
if l.valid {
|
||||
let [a, b, c, d] = c.map(|x| Point2::new(x[0], x[1]));
|
||||
l.valid = turns_left([&a?, &b?, &c?, &d?]);
|
||||
}
|
||||
}
|
||||
Ok(l)
|
||||
}
|
||||
|
||||
/// Nodes whose quadrilaterals score below this get optimization-based smoothing.
|
||||
const POOR: f64 = 0.3;
|
||||
/// Compass-search iterations per node.
|
||||
const SEARCH_STEPS: usize = 48;
|
||||
|
||||
/// Compass search for the best position of `v` ([`Local::key`]: untangle first, then maximize
|
||||
/// the smallest scaled Jacobian). Returns the new position when it improves.
|
||||
fn optimize(nodes: &[[f64; 2]], quads: &[[u32; 4]], incident: &[u32], neighbours: &[u32], v: u32) -> Result<Option<[f64; 2]>> {
|
||||
let old = nodes.get(v as usize).copied().ok_or_else(|| Error::Failed("node index".into()))?;
|
||||
let start = local(nodes, quads, incident, v, old)?;
|
||||
if start.valid && start.sj >= POOR {
|
||||
return Ok(None);
|
||||
}
|
||||
let mut reach = 0.0_f64;
|
||||
for &u in neighbours {
|
||||
let p = nodes.get(u as usize).copied().ok_or_else(|| Error::Failed("node index".into()))?;
|
||||
reach = reach.max(((p[0] - old[0]).powi(2) + (p[1] - old[1]).powi(2)).sqrt());
|
||||
}
|
||||
let (mut at, mut best) = (old, start);
|
||||
let mut step = 0.25 * reach;
|
||||
let floor = 1e-4 * step;
|
||||
const D: f64 = std::f64::consts::FRAC_1_SQRT_2;
|
||||
const DIRS: [[f64; 2]; 8] = [[1.0, 0.0], [0.0, 1.0], [-1.0, 0.0], [0.0, -1.0], [D, D], [-D, D], [-D, -D], [D, -D]];
|
||||
for _ in 0..SEARCH_STEPS {
|
||||
if !(step > floor) {
|
||||
break;
|
||||
}
|
||||
let mut moved = false;
|
||||
for d in DIRS {
|
||||
let to = [snap_to_domain(at[0] + step * d[0]), snap_to_domain(at[1] + step * d[1])];
|
||||
let l = local(nodes, quads, incident, v, to)?;
|
||||
if l.better(&best) {
|
||||
(at, best) = (to, l);
|
||||
moved = true;
|
||||
break;
|
||||
}
|
||||
}
|
||||
if !moved {
|
||||
step *= 0.5;
|
||||
}
|
||||
}
|
||||
Ok((at != old && best.better(&start)).then_some(at))
|
||||
}
|
||||
|
||||
/// Smooth the nodes from `first_free` on for at most `sweeps` sweeps. Returns the sweeps run and
|
||||
/// whether the largest move of the last sweep fell below `1e-4` of the mean edge length.
|
||||
pub(crate) fn smooth(nodes: &mut [[f64; 2]], quads: &[[u32; 4]], first_free: usize, sweeps: usize) -> Result<(usize, bool)> {
|
||||
let n = nodes.len();
|
||||
let mut edges = Vec::with_capacity(8 * quads.len());
|
||||
let mut inc = Vec::with_capacity(4 * quads.len());
|
||||
let mut total = 0.0;
|
||||
for (e, q) in quads.iter().enumerate() {
|
||||
for k in 0..4 {
|
||||
let (a, b) = (q[k], q[(k + 1) % 4]);
|
||||
edges.push((a, b));
|
||||
edges.push((b, a));
|
||||
inc.push((a, e as u32));
|
||||
if let (Some(pa), Some(pb)) = (nodes.get(a as usize), nodes.get(b as usize)) {
|
||||
total += ((pa[0] - pb[0]).powi(2) + (pa[1] - pb[1]).powi(2)).sqrt();
|
||||
}
|
||||
}
|
||||
}
|
||||
if quads.is_empty() {
|
||||
return Ok((0, true));
|
||||
}
|
||||
let tolerance = 1e-4 * total / (4 * quads.len()) as f64;
|
||||
let neighbours = Csr::of(n, &mut edges);
|
||||
let incident = Csr::of(n, &mut inc);
|
||||
for sweep in 0..sweeps {
|
||||
let mut largest = 0.0_f64;
|
||||
for v in first_free..n {
|
||||
let nb = neighbours.get(v);
|
||||
if nb.is_empty() {
|
||||
continue;
|
||||
}
|
||||
let old = nodes.get(v).copied().ok_or_else(|| Error::Failed("node index".into()))?;
|
||||
let mut avg = [0.0, 0.0];
|
||||
for &u in nb {
|
||||
let p = nodes.get(u as usize).copied().ok_or_else(|| Error::Failed("node index".into()))?;
|
||||
avg = [avg[0] + p[0], avg[1] + p[1]];
|
||||
}
|
||||
let avg = [avg[0] / nb.len() as f64, avg[1] / nb.len() as f64];
|
||||
let quads_of_v = incident.get(v);
|
||||
let before = local(nodes, quads, quads_of_v, v as u32, old)?;
|
||||
for step in [1.0, 0.5, 0.25] {
|
||||
let at = [snap_to_domain(old[0] + step * (avg[0] - old[0])), snap_to_domain(old[1] + step * (avg[1] - old[1]))];
|
||||
if at == old {
|
||||
break;
|
||||
}
|
||||
if local(nodes, quads, quads_of_v, v as u32, at)?.no_worse(&before) {
|
||||
largest = largest.max(((at[0] - old[0]).powi(2) + (at[1] - old[1]).powi(2)).sqrt());
|
||||
if let Some(x) = nodes.get_mut(v) {
|
||||
*x = at;
|
||||
}
|
||||
break;
|
||||
}
|
||||
}
|
||||
}
|
||||
if largest <= tolerance {
|
||||
let polished = polish(nodes, quads, first_free, &neighbours, &incident, sweeps)?;
|
||||
return Ok((sweep + 1, polished));
|
||||
}
|
||||
}
|
||||
polish(nodes, quads, first_free, &neighbours, &incident, sweeps)?;
|
||||
Ok((sweeps, false))
|
||||
}
|
||||
|
||||
/// Optimization-based smoothing of the poor and inverted nodes, at most `sweeps` sweeps.
|
||||
/// Returns whether a sweep ended with nothing left to improve.
|
||||
fn polish(nodes: &mut [[f64; 2]], quads: &[[u32; 4]], first_free: usize, neighbours: &Csr, incident: &Csr, sweeps: usize) -> Result<bool> {
|
||||
for _ in 0..sweeps {
|
||||
let mut changed = false;
|
||||
for v in first_free..nodes.len() {
|
||||
if let Some(at) = optimize(nodes, quads, incident.get(v), neighbours.get(v), v as u32)? {
|
||||
if let Some(x) = nodes.get_mut(v) {
|
||||
*x = at;
|
||||
}
|
||||
changed = true;
|
||||
}
|
||||
}
|
||||
if !changed {
|
||||
return Ok(true);
|
||||
}
|
||||
}
|
||||
Ok(false)
|
||||
}
|
||||
@@ -0,0 +1,539 @@
|
||||
use std::collections::BTreeMap;
|
||||
use std::f64::consts::PI;
|
||||
|
||||
use super::*;
|
||||
use crate::quality::Validity;
|
||||
use crate::size::{UniformSize, divide_polyline, even_divisions, ideal_divisions, loop_divisions};
|
||||
|
||||
fn opts() -> QuadOptions {
|
||||
QuadOptions::default()
|
||||
}
|
||||
|
||||
/// A `w × h` rectangle at `(x0, y0)`, counterclockwise, `nx` divisions along x and `ny` along y.
|
||||
fn rect(x0: f64, y0: f64, w: f64, h: f64, nx: usize, ny: usize) -> Vec<[f64; 2]> {
|
||||
let mut p = Vec::new();
|
||||
for i in 0..nx {
|
||||
p.push([x0 + w * i as f64 / nx as f64, y0]);
|
||||
}
|
||||
for j in 0..ny {
|
||||
p.push([x0 + w, y0 + h * j as f64 / ny as f64]);
|
||||
}
|
||||
for i in 0..nx {
|
||||
p.push([x0 + w - w * i as f64 / nx as f64, y0 + h]);
|
||||
}
|
||||
for j in 0..ny {
|
||||
p.push([x0, y0 + h - h * j as f64 / ny as f64]);
|
||||
}
|
||||
p
|
||||
}
|
||||
|
||||
fn circle(cx: f64, cy: f64, r: f64, n: usize) -> Vec<[f64; 2]> {
|
||||
(0..n).map(|k| 2.0 * PI * k as f64 / n as f64).map(|t| [cx + r * t.cos(), cy + r * t.sin()]).collect()
|
||||
}
|
||||
|
||||
/// A closed polygon with every side divided evenly into `per_side` pieces.
|
||||
fn subdivide(corners: &[[f64; 2]], per_side: usize) -> Vec<[f64; 2]> {
|
||||
let mut p = Vec::new();
|
||||
for (i, a) in corners.iter().enumerate() {
|
||||
let b = corners[(i + 1) % corners.len()];
|
||||
for k in 0..per_side {
|
||||
let t = k as f64 / per_side as f64;
|
||||
p.push([a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t]);
|
||||
}
|
||||
}
|
||||
p
|
||||
}
|
||||
|
||||
fn signed_area(p: &[[f64; 2]]) -> f64 {
|
||||
(0..p.len()).map(|i| p[i][0] * p[(i + 1) % p.len()][1] - p[(i + 1) % p.len()][0] * p[i][1]).sum::<f64>() / 2.0
|
||||
}
|
||||
|
||||
/// Every mesh invariant: boundary nodes unchanged and first, every element valid by the exact
|
||||
/// test and counterclockwise, conformity (each interior edge used once in each direction, each
|
||||
/// boundary edge once), the boundary edges exactly the input loops, and the area.
|
||||
fn check(mesh: &QuadMesh, loops: &[Vec<[f64; 2]>]) {
|
||||
let input: Vec<[f64; 2]> = loops.iter().flatten().copied().collect();
|
||||
assert_eq!(mesh.boundary_nodes, input.len());
|
||||
assert_eq!(&mesh.nodes[..input.len()], input.as_slice());
|
||||
let mut area = 0.0;
|
||||
let mut directed: BTreeMap<(u32, u32), usize> = BTreeMap::new();
|
||||
for q in &mesh.quads {
|
||||
let c = q.map(|v| mesh.nodes[v as usize]);
|
||||
assert_eq!(crate::quality::planar_quad_validity(&c).unwrap(), Validity::Valid, "{c:?}");
|
||||
area += signed_area(&c);
|
||||
for k in 0..4 {
|
||||
*directed.entry((q[k], q[(k + 1) % 4])).or_default() += 1;
|
||||
}
|
||||
}
|
||||
assert!(directed.values().all(|&n| n == 1), "an edge is used twice in the same direction");
|
||||
let boundary: Vec<(u32, u32)> = directed.keys().filter(|(a, b)| !directed.contains_key(&(*b, *a))).copied().collect();
|
||||
let mut expected = Vec::new();
|
||||
let mut base = 0u32;
|
||||
for l in loops {
|
||||
let n = l.len() as u32;
|
||||
let ccw = signed_area(l) > 0.0;
|
||||
let outer = base == 0;
|
||||
for i in 0..n {
|
||||
let (a, b) = (base + i, base + (i + 1) % n);
|
||||
// The outer loop runs counterclockwise around the mesh, holes clockwise.
|
||||
expected.push(if ccw == outer { (a, b) } else { (b, a) });
|
||||
}
|
||||
base += n;
|
||||
}
|
||||
expected.sort_unstable();
|
||||
assert_eq!(boundary, expected, "the mesh boundary is not the input loops");
|
||||
let want: f64 = loops.iter().enumerate().map(|(k, l)| if k == 0 { signed_area(l).abs() } else { -signed_area(l).abs() }).sum();
|
||||
assert!((area - want).abs() <= 1e-9 * want.abs().max(1.0), "area {area} vs {want}");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn rectangle_free() {
|
||||
let loops = vec![rect(0.0, 0.0, 10.0, 5.0, 20, 10)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert!(m.stats.min_scaled_jacobian > 0.3, "{:?}", m.stats);
|
||||
// About (10 × 5) / 0.5² = 200 quads.
|
||||
assert!((100..400).contains(&m.quads.len()), "{}", m.quads.len());
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn single_quad_and_a_six_point_loop() {
|
||||
let quad = vec![vec![[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.0, 1.0]]];
|
||||
let m = free_quad_mesh(&quad, &UniformSize(1.0), &opts()).unwrap();
|
||||
assert_eq!(m.quads.len(), 1);
|
||||
check(&m, &quad);
|
||||
let six = vec![rect(0.0, 0.0, 2.0, 1.0, 2, 1)];
|
||||
let m = free_quad_mesh(&six, &UniformSize(1.0), &opts()).unwrap();
|
||||
check(&m, &six);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn clockwise_input_is_accepted() {
|
||||
let mut l = rect(0.0, 0.0, 4.0, 2.0, 8, 4);
|
||||
l.reverse();
|
||||
let loops = vec![l];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn l_shape() {
|
||||
let corners: [[f64; 2]; 6] = [[0.0, 0.0], [10.0, 0.0], [10.0, 4.0], [4.0, 4.0], [4.0, 10.0], [0.0, 10.0]];
|
||||
// Even divisions per side keep the corners at even positions.
|
||||
let mut l = Vec::new();
|
||||
for (i, a) in corners.iter().enumerate() {
|
||||
let b = corners[(i + 1) % corners.len()];
|
||||
let len = ((b[0] - a[0]).powi(2) + (b[1] - a[1]).powi(2)).sqrt();
|
||||
let n = even_divisions(len / 0.5).unwrap();
|
||||
for k in 0..n {
|
||||
let t = k as f64 / n as f64;
|
||||
l.push([a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t]);
|
||||
}
|
||||
}
|
||||
let loops = vec![l];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert!(m.stats.min_scaled_jacobian > 0.2, "{:?}", m.stats);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn plate_with_a_hole() {
|
||||
let loops = vec![rect(0.0, 0.0, 20.0, 10.0, 40, 20), circle(10.0, 5.0, 2.5, 32)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert!(m.stats.min_scaled_jacobian > 0.2, "{:?}", m.stats);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn plate_with_many_holes() {
|
||||
let mut loops = vec![rect(0.0, 0.0, 30.0, 30.0, 30, 30)];
|
||||
for i in 0..3 {
|
||||
for j in 0..3 {
|
||||
loops.push(circle(5.0 + 10.0 * i as f64, 5.0 + 10.0 * j as f64, 2.0 + 0.5 * ((i + j) % 2) as f64, 16 + 4 * i));
|
||||
}
|
||||
}
|
||||
// A square hole too, given clockwise.
|
||||
loops[5] = rect(13.0, 13.0, 4.0, 4.0, 4, 4).into_iter().rev().collect();
|
||||
let m = free_quad_mesh(&loops, &UniformSize(1.0), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn thin_sliver() {
|
||||
let loops = vec![rect(0.0, 0.0, 100.0, 1.0, 200, 2)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert_eq!(m.quads.len(), 400);
|
||||
// A sliver one division across: the split puts a mid node on each short side.
|
||||
let loops = vec![rect(0.0, 0.0, 50.0, 0.25, 100, 1)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
fn star(r_out: f64, r_in: f64) -> Vec<[f64; 2]> {
|
||||
(0..10).map(|k| (k as f64 * PI / 5.0 + PI / 2.0, if k % 2 == 0 { r_out } else { r_in })).map(|(t, r)| [r * t.cos(), r * t.sin()]).collect()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn concave_star() {
|
||||
let loops = vec![subdivide(&star(10.0, 4.0), 4)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(1.0), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
// A star with a star-shaped hole.
|
||||
let loops = vec![subdivide(&star(10.0, 4.0), 4), subdivide(&star(2.5, 1.0), 2)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(1.0), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
/// A polygon with every side divided evenly at about `h` (even counts, so corners stay at even
|
||||
/// positions).
|
||||
fn divided(corners: &[[f64; 2]], h: f64) -> Vec<[f64; 2]> {
|
||||
let mut l = Vec::new();
|
||||
for (i, a) in corners.iter().enumerate() {
|
||||
let b = corners[(i + 1) % corners.len()];
|
||||
let n = even_divisions(((b[0] - a[0]).powi(2) + (b[1] - a[1]).powi(2)).sqrt() / h).unwrap();
|
||||
l.extend((0..n).map(|k| k as f64 / n as f64).map(|t| [a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t]));
|
||||
}
|
||||
l
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn i_beam_end_face() {
|
||||
// Flanges 10 × 1, web 1 thick, 12 deep overall: thin walls meeting at reflex corners.
|
||||
let c = [
|
||||
[0.0, 0.0],
|
||||
[10.0, 0.0],
|
||||
[10.0, 1.0],
|
||||
[5.5, 1.0],
|
||||
[5.5, 11.0],
|
||||
[10.0, 11.0],
|
||||
[10.0, 12.0],
|
||||
[0.0, 12.0],
|
||||
[0.0, 11.0],
|
||||
[4.5, 11.0],
|
||||
[4.5, 1.0],
|
||||
[0.0, 1.0],
|
||||
];
|
||||
for h in [0.5, 0.25] {
|
||||
let loops = vec![divided(&c, h)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(h), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert!(m.stats.min_scaled_jacobian > 0.2, "{h}: {:?}", m.stats);
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn perforated_plate() {
|
||||
// A 40 × 20 plate with a 7 × 3 grid of holes of radius 1 on a 5 mm pitch.
|
||||
let h = 0.5;
|
||||
let mut loops = vec![divided(&[[0.0, 0.0], [40.0, 0.0], [40.0, 20.0], [0.0, 20.0]], h)];
|
||||
for i in 0..7 {
|
||||
for j in 0..3 {
|
||||
loops.push(circle(5.0 + 5.0 * i as f64, 5.0 + 5.0 * j as f64, 1.0, 2 * (PI / h).ceil() as usize));
|
||||
}
|
||||
}
|
||||
let m = free_quad_mesh(&loops, &UniformSize(h), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
assert!(m.stats.min_scaled_jacobian > 0.2, "{:?}", m.stats);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn comb_with_deep_slots() {
|
||||
// Teeth of width 1 between slots of width 1, 4 deep: strongly concave.
|
||||
let mut corners = vec![[0.0, 0.0], [9.0, 0.0]];
|
||||
for k in (0..5).rev() {
|
||||
let x = 2.0 * k as f64;
|
||||
corners.extend([[x + 1.0, 6.0], [x, 6.0]]);
|
||||
if k > 0 {
|
||||
corners.extend([[x, 2.0], [x - 1.0, 2.0]]);
|
||||
}
|
||||
}
|
||||
let loops = vec![subdivide(&corners, 4)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn graded_size_field() {
|
||||
let loops = vec![rect(0.0, 0.0, 20.0, 20.0, 20, 20)];
|
||||
let near = |p: [f64; 3]| 0.25 + 0.15 * (p[0] * p[0] + p[1] * p[1]).sqrt();
|
||||
let graded = free_quad_mesh(&loops, &near, &opts()).unwrap();
|
||||
check(&graded, &loops);
|
||||
let uniform = free_quad_mesh(&loops, &UniformSize(1.0), &opts()).unwrap();
|
||||
// Denser than the uniform mesh near the origin, coarser far from it.
|
||||
let within = |m: &QuadMesh, r: f64| m.nodes.iter().filter(|p| p[0] * p[0] + p[1] * p[1] < r * r).count();
|
||||
assert!(within(&graded, 8.0) > within(&uniform, 8.0), "{} {}", within(&graded, 8.0), within(&uniform, 8.0));
|
||||
assert!(graded.nodes.len() - within(&graded, 16.0) < uniform.nodes.len() - within(&uniform, 16.0));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn boundary_from_the_size_helpers() {
|
||||
// An arc-and-line boundary whose divisions come from the size field, loop parity fixed.
|
||||
let field = |p: [f64; 3]| 0.3 + 0.05 * p[0].abs();
|
||||
let arc: Vec<[f64; 3]> = (0..=32).map(|k| PI * k as f64 / 32.0).map(|t| [5.0 * t.cos(), 5.0 * t.sin(), 0.0]).collect();
|
||||
let base = vec![[-5.0, 0.0, 0.0], [5.0, 0.0, 0.0]];
|
||||
let curves = [arc, base];
|
||||
let ideal: Vec<f64> = curves.iter().map(|c| ideal_divisions(&field, c).unwrap()).collect();
|
||||
let counts = loop_divisions(&ideal).unwrap();
|
||||
let mut l = Vec::new();
|
||||
for (c, n) in curves.iter().zip(counts) {
|
||||
let pts = divide_polyline(&field, c, n).unwrap();
|
||||
l.extend(pts[..pts.len() - 1].iter().map(|p| [p[0], p[1]]));
|
||||
}
|
||||
assert_eq!(l.len() % 2, 0);
|
||||
let loops = vec![l];
|
||||
let m = free_quad_mesh(&loops, &field, &opts()).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn deterministic() {
|
||||
let loops = vec![rect(0.0, 0.0, 20.0, 10.0, 40, 20), circle(7.0, 5.0, 2.0, 24), circle(14.0, 5.0, 3.0, 30)];
|
||||
let a = free_quad_mesh(&loops, &UniformSize(0.6), &opts()).unwrap();
|
||||
let b = free_quad_mesh(&loops, &UniformSize(0.6), &opts()).unwrap();
|
||||
assert_eq!(a, b);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn no_smoothing_is_still_valid() {
|
||||
let loops = vec![rect(0.0, 0.0, 20.0, 10.0, 40, 20), circle(10.0, 5.0, 2.5, 32)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.5), &QuadOptions { smoothing_sweeps: 0, ..opts() }).unwrap();
|
||||
check(&m, &loops);
|
||||
assert_eq!(m.stats.smoothing_sweeps, 0);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn hostile_free_input() {
|
||||
let s = UniformSize(1.0);
|
||||
let ok = rect(0.0, 0.0, 4.0, 4.0, 4, 4);
|
||||
let invalid = |loops: Vec<Vec<[f64; 2]>>| matches!(free_quad_mesh(&loops, &s, &opts()), Err(Error::Invalid(_)));
|
||||
assert!(invalid(vec![]));
|
||||
assert!(invalid(vec![vec![[0.0, 0.0], [1.0, 0.0]]]));
|
||||
// Odd count: an error, never a silent repair.
|
||||
let mut odd = ok.clone();
|
||||
odd.push([0.0, 0.5]);
|
||||
assert!(invalid(vec![odd]));
|
||||
let mut nan = ok.clone();
|
||||
nan[3] = [f64::NAN, 1.0];
|
||||
assert!(invalid(vec![nan]));
|
||||
let mut inf = ok.clone();
|
||||
inf[3] = [f64::INFINITY, 1.0];
|
||||
assert!(invalid(vec![inf]));
|
||||
let mut far = ok.clone();
|
||||
far[3] = [1e300, 1.0];
|
||||
assert!(invalid(vec![far]));
|
||||
// Zero-length edge and repeated point.
|
||||
let mut zero = ok.clone();
|
||||
zero[2] = zero[1];
|
||||
assert!(invalid(vec![zero]));
|
||||
let mut repeated = ok.clone();
|
||||
repeated[10] = repeated[2];
|
||||
assert!(invalid(vec![repeated]));
|
||||
// Self-intersecting (bow tie) loop.
|
||||
assert!(invalid(vec![vec![[0.0, 0.0], [2.0, 2.0], [2.0, 1.0], [2.0, 0.0], [0.0, 2.0], [0.0, 1.0]]]));
|
||||
// A point touching another edge.
|
||||
assert!(invalid(vec![vec![[0.0, 0.0], [4.0, 0.0], [4.0, 4.0], [2.0, 0.0], [0.0, 4.0], [0.0, 2.0]]]));
|
||||
// Collinear fold-back.
|
||||
assert!(invalid(vec![vec![[0.0, 0.0], [4.0, 0.0], [2.0, 0.0], [2.0, 2.0], [1.0, 2.0], [0.0, 2.0]]]));
|
||||
// All collinear.
|
||||
assert!(invalid(vec![vec![[0.0, 0.0], [1.0, 0.0], [2.0, 0.0], [3.0, 0.0], [4.0, 0.0], [5.0, 0.0]]]));
|
||||
// Holes: outside, crossing the outer loop, nested, too few points.
|
||||
assert!(invalid(vec![ok.clone(), circle(10.0, 10.0, 1.0, 8)]));
|
||||
assert!(invalid(vec![ok.clone(), circle(4.0, 2.0, 1.0, 8)]));
|
||||
assert!(invalid(vec![rect(0.0, 0.0, 10.0, 10.0, 10, 10), circle(5.0, 5.0, 3.0, 16), circle(5.0, 5.0, 1.0, 8)]));
|
||||
assert!(invalid(vec![ok.clone(), rect(1.0, 1.0, 2.0, 2.0, 1, 1)]));
|
||||
// Two holes crossing each other.
|
||||
assert!(invalid(vec![rect(0.0, 0.0, 10.0, 10.0, 10, 10), circle(4.0, 5.0, 2.0, 16), circle(6.0, 5.0, 2.0, 16)]));
|
||||
// Bad size fields.
|
||||
for h in [0.0, -1.0, f64::NAN, f64::INFINITY] {
|
||||
assert!(matches!(free_quad_mesh(std::slice::from_ref(&ok), &UniformSize(h), &opts()), Err(Error::Invalid(_))), "{h}");
|
||||
}
|
||||
// Limits.
|
||||
assert!(matches!(free_quad_mesh(std::slice::from_ref(&ok), &UniformSize(1e-4), &opts()), Err(Error::Limit(_))));
|
||||
assert!(matches!(free_quad_mesh(&[ok], &UniformSize(0.1), &QuadOptions { max_elements: 10, ..opts() }), Err(Error::Limit(_))));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn random_boundaries_never_panic() {
|
||||
// Pseudo-random jagged loops (a fixed linear congruential sequence), every point a corner:
|
||||
// every result is a valid mesh or an input error.
|
||||
let mut seed = 12345u64;
|
||||
let mut next = || {
|
||||
seed = seed.wrapping_mul(6364136223846793005).wrapping_add(1442695040888963407);
|
||||
((seed >> 33) as f64) / (1u64 << 31) as f64
|
||||
};
|
||||
let (mut meshed, mut rejected) = (0, 0);
|
||||
for case in 0..200 {
|
||||
let n = 6 + 2 * (case % 10);
|
||||
let mut loops = vec![
|
||||
(0..n)
|
||||
.map(|k| {
|
||||
let t = 2.0 * PI * k as f64 / n as f64;
|
||||
let r = 1.0 + 3.0 * next();
|
||||
[r * t.cos(), r * t.sin()]
|
||||
})
|
||||
.collect::<Vec<_>>(),
|
||||
];
|
||||
if case % 3 == 0 {
|
||||
loops.push(circle(0.0, 0.0, 0.5 * next(), 6));
|
||||
}
|
||||
if case % 7 == 0 {
|
||||
loops[0] = (0..n).map(|_| [10.0 * next() - 5.0, 10.0 * next() - 5.0]).collect();
|
||||
}
|
||||
match free_quad_mesh(&loops, &UniformSize(0.2 + next()), &opts()) {
|
||||
Ok(m) => {
|
||||
check(&m, &loops);
|
||||
meshed += 1;
|
||||
}
|
||||
Err(Error::Invalid(_)) => rejected += 1,
|
||||
Err(e) => panic!("case {case}: {e}"),
|
||||
}
|
||||
}
|
||||
// Every valid region is meshed; the rest are self-intersecting loops.
|
||||
assert_eq!(meshed + rejected, 200);
|
||||
assert!(meshed > 150, "{meshed}");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn mapped_rectangle_is_the_grid() {
|
||||
let l = rect(0.0, 0.0, 4.0, 2.0, 4, 2);
|
||||
let m = mapped_quad_mesh(&l, [0, 4, 6, 10], &opts()).unwrap();
|
||||
assert_eq!((m.nu, m.nv), (4, 2));
|
||||
assert_eq!(m.mesh.quads.len(), 8);
|
||||
check(&m.mesh, std::slice::from_ref(&l));
|
||||
for j in 0..=2 {
|
||||
for i in 0..=4 {
|
||||
assert_eq!(m.mesh.nodes[m.grid[j * 5 + i] as usize], [i as f64, j as f64]);
|
||||
}
|
||||
}
|
||||
assert!((m.mesh.stats.min_scaled_jacobian - 1.0).abs() < 1e-12);
|
||||
assert!(m.mesh.stats.max_skew < 1e-12);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn mapped_curved_and_graded() {
|
||||
// A quarter annulus, radii 1 and 2, graded radially.
|
||||
let (na, nr) = (8, 5);
|
||||
let r = |j: usize| 1.0 + (j as f64 / nr as f64).powi(2);
|
||||
let mut l = Vec::new();
|
||||
for j in 0..nr {
|
||||
l.push([r(j), 0.0]);
|
||||
}
|
||||
for i in 0..na {
|
||||
let t = 0.5 * PI * i as f64 / na as f64;
|
||||
l.push([2.0 * t.cos(), 2.0 * t.sin()]);
|
||||
}
|
||||
for j in 0..nr {
|
||||
l.push([0.0, r(nr - j)]);
|
||||
}
|
||||
for i in 0..na {
|
||||
let t = 0.5 * PI * (na - i) as f64 / na as f64;
|
||||
l.push([t.cos(), t.sin()]);
|
||||
}
|
||||
let corners = [0, nr, nr + na, 2 * nr + na];
|
||||
let m = mapped_quad_mesh(&l, corners, &opts()).unwrap();
|
||||
check(&m.mesh, std::slice::from_ref(&l));
|
||||
// The same region given clockwise.
|
||||
let mut cw = l.clone();
|
||||
cw.reverse();
|
||||
let n = cw.len();
|
||||
let m = mapped_quad_mesh(&cw, [n - 1 - corners[3], n - 1 - corners[2], n - 1 - corners[1], n - 1 - corners[0]], &opts()).unwrap();
|
||||
check(&m.mesh, &[cw]);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn mapped_hostile() {
|
||||
let l = rect(0.0, 0.0, 4.0, 2.0, 4, 2);
|
||||
assert!(mapped_quad_mesh(&l, [0, 4, 6, 11], &opts()).is_err()); // unequal opposite sides
|
||||
assert!(mapped_quad_mesh(&l, [0, 6, 4, 10], &opts()).is_err()); // out of order
|
||||
assert!(mapped_quad_mesh(&l, [0, 4, 6, 99], &opts()).is_err());
|
||||
assert!(mapped_quad_mesh(&l, [0, 0, 6, 10], &opts()).is_err());
|
||||
let mut nan = l.clone();
|
||||
nan[5] = [f64::NAN, 0.0];
|
||||
assert!(mapped_quad_mesh(&nan, [0, 4, 6, 10], &opts()).is_err());
|
||||
// A region whose TFI folds: the top side dips almost to the bottom between its corners.
|
||||
let fold = vec![[0.0, 0.0], [1.0, 0.0], [2.0, 0.0], [3.0, 0.0], [4.0, 0.0], [4.0, 4.0], [3.0, 4.0], [2.2, 0.2], [1.8, 0.2], [1.0, 4.0]];
|
||||
let r = mapped_quad_mesh(&fold, [0, 4, 5, 9], &opts());
|
||||
assert!(matches!(r, Err(Error::Failed(_)) | Err(Error::Invalid(_))), "{r:?}");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn transfinite_in_space() {
|
||||
let bottom = [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [2.0, 0.0, 0.0]];
|
||||
let top = [[0.0, 1.0, 1.0], [1.0, 1.0, 1.0], [2.0, 1.0, 1.0]];
|
||||
let left = [[0.0, 0.0, 0.0], [0.0, 1.0, 1.0]];
|
||||
let right = [[2.0, 0.0, 0.0], [2.0, 1.0, 1.0]];
|
||||
let g = transfinite_grid(&bottom, &right, &top, &left).unwrap();
|
||||
assert_eq!(g.len(), 6);
|
||||
assert_eq!(g[4], [1.0, 1.0, 1.0]);
|
||||
let side = |a: [f64; 2], b: [f64; 2]| [a, [(a[0] + b[0]) / 2.0, (a[1] + b[1]) / 2.0], b];
|
||||
let g =
|
||||
transfinite_grid(&side([0.0, 0.0], [2.0, 0.0]), &side([2.0, 0.0], [2.0, 2.0]), &side([0.0, 2.0], [2.0, 2.0]), &side([0.0, 0.0], [0.0, 2.0]))
|
||||
.unwrap();
|
||||
assert_eq!(g[4], [1.0, 1.0]);
|
||||
// Corners that do not match, a zero-length side, empty sides.
|
||||
assert!(transfinite_grid(&bottom, &right, &top, &[[0.0, 0.0, 0.1], [0.0, 1.0, 1.0]]).is_err());
|
||||
assert!(transfinite_grid(&[[0.0; 2], [0.0; 2]], &[[0.0; 2], [0.0, 1.0]], &[[0.0, 1.0], [0.0, 1.0]], &[[0.0; 2], [0.0, 1.0]]).is_err());
|
||||
assert!(transfinite_grid::<2>(&[], &[], &[], &[]).is_err());
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn planar_quad_metrics() {
|
||||
use crate::quality::{Metric, planar_quad_quality, planar_quad_validity};
|
||||
let rect = [[0.0, 0.0], [2.0, 0.0], [2.0, 1.0], [0.0, 1.0]];
|
||||
let q = planar_quad_quality(&rect).unwrap();
|
||||
assert_eq!(q.validity(), Some(Validity::Valid));
|
||||
assert_eq!(q.size, 2.0);
|
||||
assert_eq!(q.get(Metric::AspectRatio), Some(2.0));
|
||||
assert_eq!(q.get(Metric::Skew), Some(0.0));
|
||||
assert_eq!(q.get(Metric::ScaledJacobian), Some(1.0));
|
||||
assert_eq!(q.get(Metric::JacobianRatio), Some(1.0));
|
||||
// A rhombus with 60° corners: its principal axes are its edge directions, so skew cos 60°
|
||||
// and aspect 1; the scaled Jacobian is sin 60°.
|
||||
let h = 3.0_f64.sqrt() / 2.0;
|
||||
let rh = [[0.0, 0.0], [1.0, 0.0], [1.5, h], [0.5, h]];
|
||||
let q = planar_quad_quality(&rh).unwrap();
|
||||
assert!((q.get(Metric::ScaledJacobian).unwrap() - h).abs() < 1e-12);
|
||||
assert!((q.get(Metric::Skew).unwrap() - 0.5).abs() < 1e-12);
|
||||
assert!((q.get(Metric::AspectRatio).unwrap() - 1.0).abs() < 1e-12);
|
||||
// Clockwise, bow tie, concave (dart), degenerate (a straight corner): invalid by the exact test.
|
||||
let cw = [rect[0], rect[3], rect[2], rect[1]];
|
||||
assert_eq!(planar_quad_validity(&cw).unwrap(), Validity::Invalid);
|
||||
assert_eq!(planar_quad_validity(&[[0.0, 0.0], [1.0, 1.0], [1.0, 0.0], [0.0, 1.0]]).unwrap(), Validity::Invalid);
|
||||
assert_eq!(planar_quad_validity(&[[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [0.7, 0.3]]).unwrap(), Validity::Invalid);
|
||||
assert_eq!(planar_quad_validity(&[[0.0, 0.0], [1.0, 0.0], [2.0, 0.0], [0.0, 1.0]]).unwrap(), Validity::Invalid);
|
||||
// A corner off by one ulp from straight is decided exactly.
|
||||
let x = 1.0 + f64::EPSILON;
|
||||
assert_eq!(planar_quad_validity(&[[0.0, 0.0], [1.0, 0.0], [2.0, x - 1.0], [0.0, 1.0]]).unwrap(), Validity::Valid);
|
||||
assert!(planar_quad_validity(&[[0.0, 0.0], [1.0, f64::NAN], [1.0, 1.0], [0.0, 1.0]]).is_err());
|
||||
let q = planar_quad_quality(&[[0.0, 0.0], [1.0, 0.0], [2.0, 0.0], [0.0, 1.0]]).unwrap();
|
||||
assert_eq!(q.get(Metric::JacobianRatio), Some(0.0));
|
||||
}
|
||||
|
||||
#[test]
|
||||
#[ignore = "timing and quality report: cargo test --release -p modelcraft-mesh quad_report -- --ignored --nocapture"]
|
||||
fn quad_report() {
|
||||
let cases: Vec<(&str, Vec<Vec<[f64; 2]>>, f64)> = vec![
|
||||
("rectangle", vec![rect(0.0, 0.0, 10.0, 5.0, 20, 10)], 0.5),
|
||||
("plate with hole", vec![rect(0.0, 0.0, 20.0, 10.0, 40, 20), circle(10.0, 5.0, 2.5, 32)], 0.5),
|
||||
("star", vec![subdivide(&star(10.0, 4.0), 4)], 1.0),
|
||||
("large plate", vec![rect(0.0, 0.0, 200.0, 100.0, 1000, 500), circle(50.0, 50.0, 20.0, 400), circle(150.0, 50.0, 20.0, 400)], 0.2),
|
||||
];
|
||||
for (name, loops, h) in cases {
|
||||
let t = std::time::Instant::now();
|
||||
let m = free_quad_mesh(&loops, &UniformSize(h), &opts()).unwrap();
|
||||
println!("{name}: {} quads in {:?}, {:?}", m.quads.len(), t.elapsed(), m.stats);
|
||||
let t = std::time::Instant::now();
|
||||
let _ = free_quad_mesh(&loops, &UniformSize(h), &QuadOptions { smoothing_sweeps: 0, ..opts() }).unwrap();
|
||||
println!(" without smoothing {:?}", t.elapsed());
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn unsmoothed_meshes_are_valid() {
|
||||
// Regression: recombination once accepted nearly straight corners that the split turned
|
||||
// degenerate after rounding the mid nodes (this plate had one before the quality floor).
|
||||
let loops = vec![rect(0.0, 0.0, 60.0, 30.0, 300, 150), circle(15.0, 15.0, 6.0, 240)];
|
||||
let m = free_quad_mesh(&loops, &UniformSize(0.2), &QuadOptions { smoothing_sweeps: 0, ..opts() }).unwrap();
|
||||
check(&m, &loops);
|
||||
}
|
||||
@@ -0,0 +1,108 @@
|
||||
//! Mapped meshing of four-sided regions by transfinite interpolation.
|
||||
//!
|
||||
//! W. J. Gordon, C. A. Hall, *Construction of curvilinear co-ordinate systems and applications to
|
||||
//! mesh generation*, IJNME 7(4):461–477, 1973: the bilinearly blended Coons map
|
||||
//! `X(u, v) = (1−v) B + v T + (1−u) L + u R − [(1−u)(1−v) P00 + u(1−v) P10 + u v P11 + (1−u) v P01]`
|
||||
//! from the four sides `B` (bottom), `R` (right), `T` (top), `L` (left) and their corners. The
|
||||
//! discrete form uses the side nodes directly; the blending parameters of node `(i, j)` follow the
|
||||
//! arc-length distribution of the sides, from the intersection of the straight parameter lines
|
||||
//! `u = (1−v) u_B(i) + v u_T(i)` and `v = (1−u) v_L(j) + u v_R(j)` (the standard treatment of
|
||||
//! unequal spacing on opposite sides, e.g. Thompson, Soni, Weatherill, *Handbook of Grid
|
||||
//! Generation*, CRC 1999, ch. 3). The map is dimension-generic: faces of a block in space use it
|
||||
//! with `D = 3`.
|
||||
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// The most nodes of one mapped grid.
|
||||
pub const MAX_GRID_NODES: usize = 1 << 24;
|
||||
|
||||
fn dist<const D: usize>(a: &[f64; D], b: &[f64; D]) -> f64 {
|
||||
a.iter().zip(b).map(|(x, y)| (x - y) * (x - y)).sum::<f64>().sqrt()
|
||||
}
|
||||
|
||||
/// Normalized cumulative arc length of a side, from 0 to 1.
|
||||
fn params<const D: usize>(side: &[[f64; D]], name: &str) -> Result<Vec<f64>> {
|
||||
let mut acc = Vec::with_capacity(side.len());
|
||||
let mut s = 0.0;
|
||||
acc.push(0.0);
|
||||
for w in side.windows(2) {
|
||||
if let [a, b] = w {
|
||||
s += dist(a, b);
|
||||
}
|
||||
acc.push(s);
|
||||
}
|
||||
if !(s > 0.0) || !s.is_finite() {
|
||||
return Err(Error::Invalid(format!("the {name} side has zero or non-finite length")));
|
||||
}
|
||||
let mut out: Vec<f64> = acc.iter().map(|x| x / s).collect();
|
||||
if let Some(l) = out.last_mut() {
|
||||
*l = 1.0;
|
||||
}
|
||||
Ok(out)
|
||||
}
|
||||
|
||||
/// The transfinite grid of a four-sided region: `(nu + 1) × (nv + 1)` points, node `(i, j)` at
|
||||
/// index `j·(nu + 1) + i`. `bottom` and `top` run in `+u` (each `nu + 1` points), `left` and
|
||||
/// `right` in `+v` (each `nv + 1` points); the corners must match exactly (`bottom[0] =
|
||||
/// left[0]`, `bottom[nu] = right[0]`, `top[0] = left[nv]`, `top[nu] = right[nv]`). Side nodes are
|
||||
/// copied exactly into the grid.
|
||||
pub fn transfinite_grid<const D: usize>(bottom: &[[f64; D]], right: &[[f64; D]], top: &[[f64; D]], left: &[[f64; D]]) -> Result<Vec<[f64; D]>> {
|
||||
let (nu, nv) = (bottom.len().saturating_sub(1), left.len().saturating_sub(1));
|
||||
if nu == 0 || nv == 0 || top.len() != bottom.len() || right.len() != left.len() {
|
||||
return Err(Error::Invalid(format!(
|
||||
"opposite sides need the same number of divisions (at least 1): bottom {nu}, top {}, left {nv}, right {}",
|
||||
top.len().saturating_sub(1),
|
||||
right.len().saturating_sub(1)
|
||||
)));
|
||||
}
|
||||
let count = (nu + 1).checked_mul(nv + 1).filter(|&c| c <= MAX_GRID_NODES);
|
||||
let count = count.ok_or_else(|| Error::Limit(format!("a {nu} × {nv} grid exceeds {MAX_GRID_NODES} nodes")))?;
|
||||
for side in [bottom, right, top, left] {
|
||||
modelcraft_model::check_coordinates(side.as_flattened())?;
|
||||
}
|
||||
let corner = |s: &[[f64; D]], last: bool| if last { s.last().copied() } else { s.first().copied() };
|
||||
let (p00, p10, p01, p11) = match (corner(bottom, false), corner(bottom, true), corner(top, false), corner(top, true)) {
|
||||
(Some(a), Some(b), Some(c), Some(d)) => (a, b, c, d),
|
||||
_ => return Err(Error::Invalid("empty side".into())),
|
||||
};
|
||||
if corner(left, false) != Some(p00) || corner(right, false) != Some(p10) || corner(left, true) != Some(p01) || corner(right, true) != Some(p11) {
|
||||
return Err(Error::Invalid("the sides of a mapped region must share their corner points".into()));
|
||||
}
|
||||
let (ub, ut, vl, vr) = (params(bottom, "bottom")?, params(top, "top")?, params(left, "left")?, params(right, "right")?);
|
||||
let mut grid = Vec::with_capacity(count);
|
||||
for j in 0..=nv {
|
||||
for i in 0..=nu {
|
||||
let (Some(&b), Some(&t), Some(&l), Some(&r)) = (bottom.get(i), top.get(i), left.get(j), right.get(j)) else {
|
||||
return Err(Error::Failed("grid index".into()));
|
||||
};
|
||||
let point = if j == 0 {
|
||||
b
|
||||
} else if j == nv {
|
||||
t
|
||||
} else if i == 0 {
|
||||
l
|
||||
} else if i == nu {
|
||||
r
|
||||
} else {
|
||||
let (u0, u1, v0, v1) = (
|
||||
ub.get(i).copied().unwrap_or(0.0),
|
||||
ut.get(i).copied().unwrap_or(0.0),
|
||||
vl.get(j).copied().unwrap_or(0.0),
|
||||
vr.get(j).copied().unwrap_or(0.0),
|
||||
);
|
||||
let (du, dv) = (u1 - u0, v1 - v0);
|
||||
// |du|, |dv| < 1 at interior nodes, so the denominator is positive.
|
||||
let u = (u0 + v0 * du) / (1.0 - du * dv);
|
||||
let v = v0 + u * dv;
|
||||
let mut x = [0.0; D];
|
||||
for (k, xk) in x.iter_mut().enumerate() {
|
||||
*xk = (1.0 - v) * b[k] + v * t[k] + (1.0 - u) * l[k] + u * r[k]
|
||||
- ((1.0 - u) * (1.0 - v) * p00[k] + u * (1.0 - v) * p10[k] + u * v * p11[k] + (1.0 - u) * v * p01[k]);
|
||||
}
|
||||
x
|
||||
};
|
||||
grid.push(point);
|
||||
}
|
||||
}
|
||||
Ok(grid)
|
||||
}
|
||||
@@ -27,8 +27,9 @@ use modelcraft_model::{ElementKind, check_coordinates};
|
||||
|
||||
pub use jacobian::{JacobianCheck, Validity};
|
||||
pub use report::{GateOptions, GateReport, MetricStats, QualityReport, enclosed_volume, quality_report, validity_gate, validity_gate_elements};
|
||||
pub(crate) use shape::{planar_corner, quad_axes};
|
||||
|
||||
use crate::predicates::{Point3, Sign, tet_volume_sign};
|
||||
use crate::predicates::{Point2, Point3, Sign, orient2d, snap_to_domain, tet_volume_sign};
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// A quality metric. Angles are in degrees.
|
||||
@@ -60,6 +61,9 @@ pub enum Metric {
|
||||
MinAngle,
|
||||
/// Largest corner angle of a triangle or quadrilateral.
|
||||
MaxAngle,
|
||||
/// Skew of a quadrilateral: `|cos θ|` of the angle between its principal axes, in `[0, 1]`;
|
||||
/// 0 for a rectangle, 1 when degenerate.
|
||||
Skew,
|
||||
}
|
||||
|
||||
impl Metric {
|
||||
@@ -76,19 +80,20 @@ impl Metric {
|
||||
Metric::EdgeRatio => "edge_ratio",
|
||||
Metric::MinAngle => "min_angle",
|
||||
Metric::MaxAngle => "max_angle",
|
||||
Metric::Skew => "skew",
|
||||
}
|
||||
}
|
||||
|
||||
/// Whether a larger value is a better element.
|
||||
pub fn higher_is_better(self) -> bool {
|
||||
!matches!(self, Metric::MaxDihedral | Metric::AspectRatio | Metric::EdgeRatio | Metric::MaxAngle)
|
||||
!matches!(self, Metric::MaxDihedral | Metric::AspectRatio | Metric::EdgeRatio | Metric::MaxAngle | Metric::Skew)
|
||||
}
|
||||
|
||||
/// The fixed histogram range; values outside it fall in the first or last bin.
|
||||
pub fn range(self) -> (f64, f64) {
|
||||
match self {
|
||||
Metric::ScaledJacobian | Metric::JacobianRatio | Metric::RadiusRatio | Metric::VolumeLength => (-1.0, 1.0),
|
||||
Metric::Shape => (0.0, 1.0),
|
||||
Metric::Shape | Metric::Skew => (0.0, 1.0),
|
||||
Metric::MinDihedral | Metric::MaxDihedral | Metric::MinAngle | Metric::MaxAngle => (0.0, 180.0),
|
||||
Metric::AspectRatio | Metric::EdgeRatio => (1.0, 11.0),
|
||||
}
|
||||
@@ -218,5 +223,52 @@ pub fn element_quality(kind: ElementKind, xyz: &[[f64; 3]]) -> Result<ElementQua
|
||||
Ok(ElementQuality { kind, size: check.volume, jacobian: Some(check), metrics })
|
||||
}
|
||||
|
||||
/// The exact validity of a planar bilinear quadrilateral with corners `p` in counterclockwise
|
||||
/// order. Its Jacobian determinant is affine in each reference coordinate (the `ξη` term of a
|
||||
/// planar bilinear map cancels), so it is positive on the whole element exactly when it is
|
||||
/// positive at the four corners, i.e. when every corner turns left: `orient2d(p[i-1], p[i],
|
||||
/// p[i+1]) > 0`, decided by the exact predicate. A zero corner (degenerate) or a negative one
|
||||
/// (concave, bow-tie or clockwise) is [`Validity::Invalid`]. Coordinates below the predicates'
|
||||
/// smallest magnitude are rounded to zero first.
|
||||
pub fn planar_quad_validity(p: &[[f64; 2]; 4]) -> Result<Validity> {
|
||||
check_coordinates(p.as_flattened())?;
|
||||
let [a, b, c, d] = p.map(|c| Point2::new(snap_to_domain(c[0]), snap_to_domain(c[1])));
|
||||
Ok(if turns_left([&a?, &b?, &c?, &d?]) { Validity::Valid } else { Validity::Invalid })
|
||||
}
|
||||
|
||||
/// The exact corner test of [`planar_quad_validity`] on admissible points: every corner of the
|
||||
/// quadrilateral turns left.
|
||||
pub(crate) fn turns_left(p: [&Point2; 4]) -> bool {
|
||||
(0..4).all(|i| orient2d(p[(i + 3) % 4], p[i], p[(i + 1) % 4]) == Sign::Positive)
|
||||
}
|
||||
|
||||
/// Every metric of a planar quadrilateral (counterclockwise corners in the xy plane) with its
|
||||
/// exact validity ([`planar_quad_validity`]): the Jacobian ratio is the smallest corner
|
||||
/// determinant over the largest magnitude (exact bounds, see there) and the size is the signed
|
||||
/// area. The shape metrics are those of [`element_quality`] for a `Quad4`.
|
||||
pub fn planar_quad_quality(p: &[[f64; 2]; 4]) -> Result<ElementQuality> {
|
||||
let validity = planar_quad_validity(p)?;
|
||||
let o = p[0];
|
||||
let l = p.map(|c| [c[0] - o[0], c[1] - o[1]]);
|
||||
let corners = [0, 1, 2, 3].map(|i| shape::planar_corner(&l, i));
|
||||
let dets = corners.map(|c| c.0);
|
||||
let lo = dets.iter().fold(f64::INFINITY, |m, &d| m.min(d));
|
||||
let hi = dets.iter().fold(0.0_f64, |m, &d| m.max(d.abs()));
|
||||
let ratio = if hi > 0.0 { (lo / hi).clamp(-1.0, 1.0) } else { 0.0 };
|
||||
let area = 0.5 * ((l[2][0] - l[0][0]) * (l[3][1] - l[1][1]) - (l[2][1] - l[0][1]) * (l[3][0] - l[1][0]));
|
||||
let lifted = l.map(|c| [c[0], c[1], 0.0]);
|
||||
let (_, mut metrics) = shape::quad(lifted);
|
||||
// Corner scaled Jacobians measured against +z (the 3D form measures against the diagonals'
|
||||
// normal, which flips with a clockwise quad).
|
||||
let scaled = corners.iter().fold(f64::INFINITY, |m, c| m.min(c.1.unwrap_or(0.0)));
|
||||
for (m, v) in metrics.iter_mut() {
|
||||
if *m == Metric::ScaledJacobian {
|
||||
*v = scaled;
|
||||
}
|
||||
}
|
||||
metrics.push((Metric::JacobianRatio, ratio));
|
||||
Ok(ElementQuality { kind: ElementKind::Quad4, size: area, jacobian: Some(JacobianCheck { validity, ratio, volume: area }), metrics })
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests;
|
||||
@@ -6,6 +6,10 @@
|
||||
//! - Hexahedra, wedges and quadrilaterals: Knupp's algebraic metrics built on the corner Jacobian
|
||||
//! matrices (Knupp, *Algebraic mesh quality metrics*, SIAM J. Sci. Comput. 23(1), 2001): the
|
||||
//! scaled Jacobian `det(A) / (|a1| |a2| |a3|)` and the shape metric `3 det(A)^{2/3} / |A|²_F`.
|
||||
//! - Quadrilateral aspect ratio and skew from the principal axes `X1 = (p1 − p0) + (p2 − p3)`,
|
||||
//! `X2 = (p2 − p1) + (p3 − p0)` (Robinson, *CRE method of element testing and the Jacobian
|
||||
//! shape parameters*, Eng. Comput. 4(2):113–118, 1987): aspect `max(|X1|, |X2|) / min(…)`, skew
|
||||
//! `|X1 · X2| / (|X1| |X2|)`.
|
||||
//!
|
||||
//! Every metric is defined for degenerate input (no division by zero, no NaN): a degenerate
|
||||
//! element scores 0 on bounded metrics and `+∞` on aspect and edge ratios.
|
||||
@@ -132,6 +136,7 @@ pub(crate) fn quad(p: [P; 4]) -> (f64, Vec<(Metric, f64)>) {
|
||||
angles = (angles.0.min(a), angles.1.max(a));
|
||||
}
|
||||
let edges = [(0, 1), (1, 2), (2, 3), (3, 0)];
|
||||
let (aspect, skew) = quad_axes(p);
|
||||
(
|
||||
area,
|
||||
vec![
|
||||
@@ -139,10 +144,36 @@ pub(crate) fn quad(p: [P; 4]) -> (f64, Vec<(Metric, f64)>) {
|
||||
(Metric::MinAngle, angles.0),
|
||||
(Metric::MaxAngle, angles.1),
|
||||
(Metric::EdgeRatio, edge_ratio(&p, &edges)),
|
||||
(Metric::AspectRatio, aspect),
|
||||
(Metric::Skew, skew),
|
||||
],
|
||||
)
|
||||
}
|
||||
|
||||
/// Corner `i` of a planar quadrilateral: the cross product `(p[i+1] − p[i]) × (p[i−1] − p[i])`
|
||||
/// (twice the corner triangle's signed area, positive for a left turn of a counterclockwise quad)
|
||||
/// and that cross over the two edge lengths (the corner's scaled Jacobian, `None` when an edge
|
||||
/// has zero length).
|
||||
pub(crate) fn planar_corner(p: &[[f64; 2]; 4], i: usize) -> (f64, Option<f64>) {
|
||||
let (a, b, c) = (p[(i + 3) % 4], p[i % 4], p[(i + 1) % 4]);
|
||||
let (u, w) = ([c[0] - b[0], c[1] - b[1]], [a[0] - b[0], a[1] - b[1]]);
|
||||
let cross = u[0] * w[1] - u[1] * w[0];
|
||||
let den = (u[0] * u[0] + u[1] * u[1]).sqrt() * (w[0] * w[0] + w[1] * w[1]).sqrt();
|
||||
let scaled = cross / den;
|
||||
(cross, (den > 0.0 && scaled.is_finite()).then_some(scaled))
|
||||
}
|
||||
|
||||
/// Robinson's aspect ratio and skew of a quadrilateral from its principal axes; `(+∞, 1)` when
|
||||
/// an axis vanishes.
|
||||
pub(crate) fn quad_axes(p: [P; 4]) -> (f64, f64) {
|
||||
let x1 = [p[1][0] - p[0][0] + p[2][0] - p[3][0], p[1][1] - p[0][1] + p[2][1] - p[3][1], p[1][2] - p[0][2] + p[2][2] - p[3][2]];
|
||||
let x2 = [p[2][0] - p[1][0] + p[3][0] - p[0][0], p[2][1] - p[1][1] + p[3][1] - p[0][1], p[2][2] - p[1][2] + p[3][2] - p[0][2]];
|
||||
let (l1, l2) = (norm(x1), norm(x2));
|
||||
let aspect = ratio(l1.max(l2), l1.min(l2), f64::INFINITY).max(1.0);
|
||||
let skew = ratio(dot(x1, x2).abs(), l1 * l2, 1.0).min(1.0);
|
||||
(aspect, skew)
|
||||
}
|
||||
|
||||
/// Wedge scaled Jacobian: the smallest determinant of the three unit edge vectors at a corner,
|
||||
/// taken in right-handed order (1 for a right prism on a right-angled corner, sin 60° ≈ 0.866 on
|
||||
/// an equilateral triangle).
|
||||
|
||||
@@ -0,0 +1,263 @@
|
||||
//! Size fields: the target element edge length at a point, and the edge divisions that follow it.
|
||||
//!
|
||||
//! One [`SizeField`] drives every mesher: the quad meshers ([`crate::quad`]) read it for interior
|
||||
//! density, and the edge-division helpers here turn it into boundary division counts so that
|
||||
//! faces sharing an edge share its nodes. The tetrahedral refinement reads the same trait.
|
||||
//!
|
||||
//! # Divisions
|
||||
//! The number of divisions of a curve `C` under a size field `h` is the length of `C` measured in
|
||||
//! units of `h`, `∫_C ds / h(s)` ([`ideal_divisions`]), rounded ([`edge_divisions`]). The nodes
|
||||
//! of a divided curve split it into pieces of equal measured length ([`divide_polyline`]), so the
|
||||
//! spacing grades with the field.
|
||||
//!
|
||||
//! # Parity
|
||||
//! An all-quadrilateral mesh of a region needs an even number of boundary edges in total (every
|
||||
//! quad has four sides, every interior edge is shared by two quads). The free quad mesher
|
||||
//! ([`crate::quad::free_quad_mesh`]) needs an even count on *every* loop. [`loop_divisions`]
|
||||
//! rounds the ideal counts of a loop's curves and fixes the parity of the loop with the change
|
||||
//! that costs the least relative spacing error; [`even_divisions`] makes a single curve even,
|
||||
//! which also keeps every loop corner at an even position (the free mesher keeps corners exactly
|
||||
//! when they are there).
|
||||
//!
|
||||
//! The integrals use the midpoint rule on [`SAMPLES_PER_SEGMENT`] pieces per polyline segment,
|
||||
//! with the size taken constant within a piece.
|
||||
//! Every function is deterministic (no transcendental functions, only `+ − × ÷ √`).
|
||||
|
||||
use crate::{Error, Result};
|
||||
|
||||
/// The target edge length at a point. Implementations return a finite, positive length; callers
|
||||
/// treat anything else as invalid input.
|
||||
pub trait SizeField {
|
||||
fn size(&self, p: [f64; 3]) -> f64;
|
||||
}
|
||||
|
||||
impl<F: Fn([f64; 3]) -> f64> SizeField for F {
|
||||
fn size(&self, p: [f64; 3]) -> f64 {
|
||||
self(p)
|
||||
}
|
||||
}
|
||||
|
||||
/// The same size everywhere.
|
||||
#[derive(Debug, Clone, Copy, PartialEq)]
|
||||
pub struct UniformSize(pub f64);
|
||||
|
||||
impl SizeField for UniformSize {
|
||||
fn size(&self, _: [f64; 3]) -> f64 {
|
||||
self.0
|
||||
}
|
||||
}
|
||||
|
||||
/// The largest division count of one curve.
|
||||
pub const MAX_DIVISIONS: usize = 1 << 20;
|
||||
/// The most points a polyline handed to the helpers may have.
|
||||
pub const MAX_POLYLINE_POINTS: usize = 1 << 20;
|
||||
/// Midpoint-rule samples per polyline segment.
|
||||
pub const SAMPLES_PER_SEGMENT: usize = 8;
|
||||
|
||||
/// The size at `p`, or an error when the field returns something that is not a finite, positive
|
||||
/// length.
|
||||
pub fn checked_size(field: &dyn SizeField, p: [f64; 3]) -> Result<f64> {
|
||||
let h = field.size(p);
|
||||
if h.is_finite() && h > 0.0 { Ok(h) } else { Err(Error::Invalid(format!("size field returned {h} at {p:?}; sizes must be finite and positive"))) }
|
||||
}
|
||||
|
||||
fn check_polyline(polyline: &[[f64; 3]]) -> Result<()> {
|
||||
if polyline.len() < 2 {
|
||||
return Err(Error::Invalid(format!("a polyline needs at least 2 points, got {}", polyline.len())));
|
||||
}
|
||||
if polyline.len() > MAX_POLYLINE_POINTS {
|
||||
return Err(Error::Limit(format!("polyline has {} points; the limit is {MAX_POLYLINE_POINTS}", polyline.len())));
|
||||
}
|
||||
modelcraft_model::check_coordinates(polyline.as_flattened())?;
|
||||
Ok(())
|
||||
}
|
||||
|
||||
fn lerp(a: [f64; 3], b: [f64; 3], t: f64) -> [f64; 3] {
|
||||
[a[0] + (b[0] - a[0]) * t, a[1] + (b[1] - a[1]) * t, a[2] + (b[2] - a[2]) * t]
|
||||
}
|
||||
|
||||
fn dist(a: [f64; 3], b: [f64; 3]) -> f64 {
|
||||
let d = [b[0] - a[0], b[1] - a[1], b[2] - a[2]];
|
||||
(d[0] * d[0] + d[1] * d[1] + d[2] * d[2]).sqrt()
|
||||
}
|
||||
|
||||
/// The polyline cut into [`SAMPLES_PER_SEGMENT`] pieces per segment, each with its measured
|
||||
/// length `∫ ds / h` (midpoint rule): `(start, end, measure)`.
|
||||
fn pieces(field: &dyn SizeField, polyline: &[[f64; 3]]) -> Result<Vec<([f64; 3], [f64; 3], f64)>> {
|
||||
check_polyline(polyline)?;
|
||||
let k = SAMPLES_PER_SEGMENT as f64;
|
||||
let mut out = Vec::with_capacity(SAMPLES_PER_SEGMENT * polyline.len());
|
||||
for w in polyline.windows(2) {
|
||||
let [a, b] = w else { continue };
|
||||
let piece = dist(*a, *b) / k;
|
||||
for s in 0..SAMPLES_PER_SEGMENT {
|
||||
let (p, q) = (lerp(*a, *b, s as f64 / k), lerp(*a, *b, (s + 1) as f64 / k));
|
||||
let m = piece / checked_size(field, lerp(*a, *b, (s as f64 + 0.5) / k))?;
|
||||
if !m.is_finite() {
|
||||
return Err(Error::Invalid("size field is too small for this polyline".into()));
|
||||
}
|
||||
out.push((p, q, m));
|
||||
}
|
||||
}
|
||||
Ok(out)
|
||||
}
|
||||
|
||||
/// The ideal (real) number of divisions of a polyline: its length measured in units of the size
|
||||
/// field.
|
||||
pub fn ideal_divisions(field: &dyn SizeField, polyline: &[[f64; 3]]) -> Result<f64> {
|
||||
let total: f64 = pieces(field, polyline)?.iter().map(|p| p.2).sum();
|
||||
if total.is_finite() { Ok(total) } else { Err(Error::Invalid("size field is too small for this polyline".into())) }
|
||||
}
|
||||
|
||||
fn rounded(ideal: f64) -> Result<usize> {
|
||||
if !(ideal >= 0.0) || !ideal.is_finite() {
|
||||
return Err(Error::Invalid(format!("ideal division count {ideal} is not a finite, non-negative number")));
|
||||
}
|
||||
if ideal > MAX_DIVISIONS as f64 {
|
||||
return Err(Error::Limit(format!("{ideal:.0} divisions exceed the limit of {MAX_DIVISIONS}")));
|
||||
}
|
||||
Ok((ideal.round() as usize).max(1))
|
||||
}
|
||||
|
||||
/// The division count of one polyline: its ideal count rounded, at least 1.
|
||||
pub fn edge_divisions(field: &dyn SizeField, polyline: &[[f64; 3]]) -> Result<usize> {
|
||||
rounded(ideal_divisions(field, polyline)?)
|
||||
}
|
||||
|
||||
/// The nearest even count to `ideal`, at least 2 (ties go up).
|
||||
pub fn even_divisions(ideal: f64) -> Result<usize> {
|
||||
let half = rounded(ideal / 2.0)?;
|
||||
half.checked_mul(2).ok_or_else(|| Error::Limit("division count overflow".into()))
|
||||
}
|
||||
|
||||
/// The relative spacing error of `n` divisions against `ideal`, as `max(n, x) / min(n, x)`.
|
||||
fn spacing_error(n: usize, ideal: f64) -> f64 {
|
||||
let (n, x) = (n as f64, ideal.max(f64::MIN_POSITIVE));
|
||||
if n > x { n / x } else { x / n }
|
||||
}
|
||||
|
||||
/// Division counts for the curves of one loop with an even total: each ideal count is rounded
|
||||
/// (at least 1); when the sum is odd, the one curve whose count changes by ±1 with the smallest
|
||||
/// increase of its spacing error is changed (ties: the lower index, then +1 before −1).
|
||||
pub fn loop_divisions(ideal: &[f64]) -> Result<Vec<usize>> {
|
||||
if ideal.is_empty() {
|
||||
return Err(Error::Invalid("a loop needs at least one curve".into()));
|
||||
}
|
||||
let mut counts = ideal.iter().map(|&x| rounded(x)).collect::<Result<Vec<_>>>()?;
|
||||
let total = counts.iter().try_fold(0usize, |s, &n| s.checked_add(n)).ok_or_else(|| Error::Limit("division count overflow".into()))?;
|
||||
if total % 2 == 0 {
|
||||
return Ok(counts);
|
||||
}
|
||||
let mut best: Option<(f64, usize, bool)> = None;
|
||||
for (i, (&n, &x)) in counts.iter().zip(ideal).enumerate() {
|
||||
let now = spacing_error(n, x);
|
||||
for up in [true, false] {
|
||||
if !up && n <= 1 {
|
||||
continue;
|
||||
}
|
||||
let m = if up { n + 1 } else { n - 1 };
|
||||
let cost = spacing_error(m, x) - now;
|
||||
if best.is_none_or(|(c, _, _)| cost < c) {
|
||||
best = Some((cost, i, up));
|
||||
}
|
||||
}
|
||||
}
|
||||
if let Some((_, i, up)) = best
|
||||
&& let Some(n) = counts.get_mut(i)
|
||||
{
|
||||
*n = if up { *n + 1 } else { *n - 1 };
|
||||
}
|
||||
Ok(counts)
|
||||
}
|
||||
|
||||
/// `n + 1` points along a polyline (the first and last are its end points, copied exactly) that
|
||||
/// split it into `n` pieces of equal length measured by the size field.
|
||||
pub fn divide_polyline(field: &dyn SizeField, polyline: &[[f64; 3]], n: usize) -> Result<Vec<[f64; 3]>> {
|
||||
if n == 0 || n > MAX_DIVISIONS {
|
||||
return Err(Error::Invalid(format!("division count {n} is outside 1..={MAX_DIVISIONS}")));
|
||||
}
|
||||
let pieces = pieces(field, polyline)?;
|
||||
let total: f64 = pieces.iter().map(|p| p.2).sum();
|
||||
let (first, last) = match (polyline.first(), polyline.last()) {
|
||||
(Some(a), Some(b)) => (*a, *b),
|
||||
_ => return Err(Error::Invalid("empty polyline".into())),
|
||||
};
|
||||
if !(total > 0.0) || !total.is_finite() {
|
||||
return Err(Error::Invalid("polyline has zero length".into()));
|
||||
}
|
||||
let mut out = Vec::with_capacity(n + 1);
|
||||
out.push(first);
|
||||
let (mut piece, mut before) = (0usize, 0.0);
|
||||
for k in 1..n {
|
||||
let target = total * k as f64 / n as f64;
|
||||
while let Some(&(_, _, m)) = pieces.get(piece) {
|
||||
if before + m >= target || piece + 1 == pieces.len() {
|
||||
break;
|
||||
}
|
||||
before += m;
|
||||
piece += 1;
|
||||
}
|
||||
let (a, b, m) = pieces.get(piece).copied().ok_or_else(|| Error::Failed("polyline division walked off the end".into()))?;
|
||||
// The size is taken constant within a piece.
|
||||
let t = if m > 0.0 { ((target - before) / m).clamp(0.0, 1.0) } else { 0.0 };
|
||||
out.push(lerp(a, b, t));
|
||||
}
|
||||
out.push(last);
|
||||
Ok(out)
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
|
||||
#[test]
|
||||
fn uniform_divisions() {
|
||||
let line = [[0.0, 0.0, 0.0], [10.0, 0.0, 0.0]];
|
||||
assert_eq!(edge_divisions(&UniformSize(1.0), &line).unwrap(), 10);
|
||||
assert_eq!(edge_divisions(&UniformSize(3.0), &line).unwrap(), 3);
|
||||
assert_eq!(edge_divisions(&UniformSize(100.0), &line).unwrap(), 1);
|
||||
let pts = divide_polyline(&UniformSize(1.0), &line, 4).unwrap();
|
||||
assert_eq!(pts, vec![[0.0, 0.0, 0.0], [2.5, 0.0, 0.0], [5.0, 0.0, 0.0], [7.5, 0.0, 0.0], [10.0, 0.0, 0.0]]);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn graded_division_is_finer_where_the_size_is_smaller() {
|
||||
let field = |p: [f64; 3]| 0.5 + 0.1 * p[0];
|
||||
let line = [[0.0, 0.0, 0.0], [5.0, 0.0, 0.0], [10.0, 0.0, 0.0]];
|
||||
let pts = divide_polyline(&field, &line, 6).unwrap();
|
||||
let gaps: Vec<f64> = pts.windows(2).map(|w| w[1][0] - w[0][0]).collect();
|
||||
assert!(gaps.windows(2).all(|g| g[1] > g[0]), "{gaps:?}");
|
||||
assert_eq!(pts.last(), Some(&[10.0, 0.0, 0.0]));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn parity() {
|
||||
assert_eq!(even_divisions(3.2).unwrap(), 4);
|
||||
assert_eq!(even_divisions(0.1).unwrap(), 2);
|
||||
assert_eq!(even_divisions(5.0).unwrap(), 6);
|
||||
let c = loop_divisions(&[3.0, 2.0, 3.0, 2.0]).unwrap();
|
||||
assert_eq!(c, vec![3, 2, 3, 2]);
|
||||
let c = loop_divisions(&[3.0, 2.0, 3.0, 2.6]).unwrap();
|
||||
assert_eq!(c.iter().sum::<usize>() % 2, 0);
|
||||
// The curve whose rounding was the worst absorbs the fix.
|
||||
assert_eq!(c, vec![3, 2, 3, 2]);
|
||||
let c = loop_divisions(&[1.0, 1.0, 1.0]).unwrap();
|
||||
assert_eq!(c, vec![2, 1, 1]);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn hostile() {
|
||||
let line = [[0.0, 0.0, 0.0], [1.0, 0.0, 0.0]];
|
||||
assert!(edge_divisions(&UniformSize(0.0), &line).is_err());
|
||||
assert!(edge_divisions(&UniformSize(f64::NAN), &line).is_err());
|
||||
assert!(edge_divisions(&UniformSize(-1.0), &line).is_err());
|
||||
assert!(edge_divisions(&UniformSize(1e-300), &line).is_err());
|
||||
assert!(edge_divisions(&UniformSize(1.0), &[[0.0; 3]]).is_err());
|
||||
assert!(edge_divisions(&UniformSize(1.0), &[[0.0; 3], [f64::NAN, 0.0, 0.0]]).is_err());
|
||||
assert!(divide_polyline(&UniformSize(1.0), &[[0.0; 3], [0.0; 3]], 2).is_err());
|
||||
assert!(divide_polyline(&UniformSize(1.0), &line, 0).is_err());
|
||||
assert!(loop_divisions(&[]).is_err());
|
||||
assert!(loop_divisions(&[f64::INFINITY]).is_err());
|
||||
assert!(even_divisions(f64::NAN).is_err());
|
||||
}
|
||||
}
|
||||
@@ -68,7 +68,14 @@ Rules beyond the layer order:
|
||||
recombination (Pellerin et al. 2018), then polycube (DualCube labelling) for blocky parts;
|
||||
frame-field/integer-grid-map methods stay research. Validity uses a certified trilinear Jacobian
|
||||
check (Marschner et al. 2020), not corner Jacobians alone.
|
||||
5. **Quality.** Scaled Jacobian, aspect ratio, dihedral angles; every generated mesh is checked
|
||||
5. **Structured hex and face meshing.** Faces get all-quad meshes on shared, prescribed edge
|
||||
nodes (`modelcraft-mesh::quad`): transfinite interpolation for four-sided faces (Gordon and
|
||||
Hall 1973), and for free faces a half-resolution constrained Delaunay triangulation,
|
||||
recombination (Lee and Lo 1994) and one uniform split that lands on the boundary nodes, then
|
||||
smoothing and untangling; every quad is checked with exact predicates. One `SizeField` trait
|
||||
(`modelcraft-mesh::size`) sets interior sizes and edge divisions, with even counts per loop.
|
||||
Mapped and swept hex build on these face meshes.
|
||||
6. **Quality.** Scaled Jacobian, aspect ratio, dihedral angles; every generated mesh is checked
|
||||
for positive Jacobians and conserved volume before it reaches the model.
|
||||
|
||||
## Commands
|
||||
|
||||
@@ -783,6 +783,73 @@ Delaunay methods with Voronoi/ODT improvement, then (2) all-hex and hex-dominant
|
||||
- **Knupp 2001.** *Algebraic mesh quality metrics.* SIAM J. Sci. Comput. 23(1):193–218. Corner-Jacobian scaled Jacobian and
|
||||
shape metrics for hexes, wedges and quads.
|
||||
|
||||
|
||||
### B7. Planar quadrilateral meshing (implemented in `modelcraft-mesh::quad`)
|
||||
|
||||
The face meshers under mapped and swept hex meshing and shell meshing. All quads on prescribed
|
||||
boundary nodes, so faces sharing an edge share its nodes. DOIs below were checked through
|
||||
`doi.org/api/handles` and Crossref on 2026-10-13; papers without a DOI are cited by venue only.
|
||||
|
||||
**B7.1 Gordon and Hall 1973, transfinite interpolation.** **Verdict: BUILD ON (mapped meshing).**
|
||||
- **Citation.** W. J. Gordon, C. A. Hall. *Construction of curvilinear co-ordinate systems and applications to mesh
|
||||
generation.* IJNME 7(4):461–477, 1973. DOI [10.1002/nme.1620070405](https://doi.org/10.1002/nme.1620070405).
|
||||
- **Use.** The bilinearly blended Coons map of a four-sided region, dimension-generic (`quad::transfinite_grid`),
|
||||
with arc-length blending parameters for unequal spacing on opposite sides.
|
||||
|
||||
**B7.2 Free all-quad: the methods compared.**
|
||||
- **Paving.** T. D. Blacker, M. B. Stephenson. *Paving: a new approach to automated quadrilateral mesh generation.*
|
||||
IJNME 32(4):811–847, 1991. DOI [10.1002/nme.1620320410](https://doi.org/10.1002/nme.1620320410). **Verdict:
|
||||
reference.** Best boundary rows; front collisions are its failure mode.
|
||||
- **Q-Morph.** S. J. Owen, M. L. Staten, S. A. Canann, S. Saigal. *Q-Morph: an indirect approach to advancing front
|
||||
quad meshing.* IJNME 44(9):1317–1340, 1999. DOI
|
||||
[10.1002/(SICI)1097-0207(19990330)44:9<1317::AID-NME532>3.0.CO;2-N](https://doi.org/10.1002/(SICI)1097-0207(19990330)44:9%3C1317::AID-NME532%3E3.0.CO;2-N).
|
||||
**Verdict: reference (upgrade path for boundary alignment).** Front states, side-edge recovery by swaps, seam
|
||||
closing.
|
||||
- **Blossom-Quad.** J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen, C. Geuzaine. *Blossom-Quad: a
|
||||
non-uniform quadrilateral mesh generator using a minimum-cost perfect-matching algorithm.* IJNME 89(9):1102–1119,
|
||||
2012. DOI [10.1002/nme.3279](https://doi.org/10.1002/nme.3279). **Verdict: reference (upgrade path for the
|
||||
matching).** Paper only: the implementation lives in Gmsh, which is GPL and forbidden.
|
||||
- **Recombination and split.** D. T. Lee, S. H. Lo. *A new scheme for the generation of a graded quadrilateral
|
||||
mesh.* Computers & Structures 52(5):847–857, 1994. DOI
|
||||
[10.1016/0045-7949(94)90070-1](https://doi.org/10.1016/0045-7949(94)90070-1). **Verdict: BUILD ON.** Triangles
|
||||
merged into quads, then every element split once, which makes the mesh all-quad whatever the merge found.
|
||||
The split topology is that of E. Catmull, J. Clark, CAD 10(6):350–355, 1978, DOI
|
||||
[10.1016/0010-4485(78)90110-0](https://doi.org/10.1016/0010-4485(78)90110-0).
|
||||
- **Why the split method.** All-quad by construction given an even point count per loop; robustness is that of the
|
||||
constrained Delaunay triangulation (exists for every valid input; exact predicates); low implementation risk.
|
||||
Its cost is irregular nodes and 120° corners in triangle-born quads, reduced by smoothing. The reasoning in full
|
||||
is in the `quad` module docs.
|
||||
|
||||
**B7.3 Planar constrained Delaunay triangulation and refinement.** **Verdict: BUILD ON.**
|
||||
- A. Bowyer, *Computing Dirichlet tessellations*, Comput. J. 24(2):162–166, 1981,
|
||||
DOI [10.1093/comjnl/24.2.162](https://doi.org/10.1093/comjnl/24.2.162); D. F. Watson, *Computing the n-dimensional
|
||||
Delaunay tessellation…*, Comput. J. 24(2):167–172, 1981, DOI
|
||||
[10.1093/comjnl/24.2.167](https://doi.org/10.1093/comjnl/24.2.167).
|
||||
- S. W. Sloan. *A fast algorithm for generating constrained Delaunay triangulations.* Computers & Structures
|
||||
47(3):441–450, 1993. DOI [10.1016/0045-7949(93)90239-A](https://doi.org/10.1016/0045-7949(93)90239-A). Edge
|
||||
recovery by flipping.
|
||||
- J. Ruppert. *A Delaunay refinement algorithm for quality 2-dimensional mesh generation.* J. Algorithms
|
||||
18(3):548–585, 1995. DOI [10.1006/jagm.1995.1021](https://doi.org/10.1006/jagm.1995.1021); J. R. Shewchuk,
|
||||
*Delaunay refinement algorithms for triangular mesh generation*, CGTA 22:21–74, 2002, DOI
|
||||
[10.1016/S0925-7721(01)00047-5](https://doi.org/10.1016/S0925-7721(01)00047-5). Used without segment splitting
|
||||
(boundary nodes are prescribed).
|
||||
|
||||
**B7.4 Smoothing and untangling of the quads.** **Verdict: BUILD ON.**
|
||||
- L. A. Freitag. *On combining Laplacian and optimization-based mesh smoothing techniques.* AMD-Vol. 220, ASME 1997;
|
||||
S. A. Canann, J. R. Tristano, M. L. Staten, *An approach to combined Laplacian and optimization-based smoothing for
|
||||
triangular, quadrilateral, and quad-dominant meshes*, IMR 1998. Laplacian moves kept only when the worst element
|
||||
does not get worse.
|
||||
- L. A. Freitag, P. Plassmann. *Local optimization-based simplicial mesh untangling and improvement.* IJNME
|
||||
49(1–2):109–125, 2000. DOI
|
||||
[10.1002/1097-0207(20000910/20)49:1/2<109::AID-NME925>3.0.CO;2-U](https://doi.org/10.1002/1097-0207(20000910/20)49:1/2%3C109::AID-NME925%3E3.0.CO;2-U).
|
||||
Untangling by maximizing the smallest (linear) corner area.
|
||||
|
||||
**B7.5 Quadrilateral quality.** **Verdict: BUILD ON.**
|
||||
- Knupp 2001 (B6.6) for the scaled Jacobian. J. Robinson, *CRE method of element testing and the Jacobian shape
|
||||
parameters*, Eng. Comput. 4(2):113–118, 1987, for aspect ratio and skew from the principal axes. Planar validity
|
||||
is exact: a planar bilinear quad's Jacobian is affine in each reference coordinate, so four exact `orient2d` corner
|
||||
tests decide it.
|
||||
|
||||
---
|
||||
|
||||
## Part C: Datasets and benchmarks
|
||||
|
||||
Reference in new issue
Block a user