rtx-cfd: indexed polygon SDF — bit-identical queries, the fluid's measured hot function cut

The 2026-08-30 fluid profile (symbolized samples, rigid AND coupled
phases of the FSI3 default) attributed the fluid step to the function:
polygon_signed_distance 51% rigid / 35% coupled — the embedded mask
rebuild and its ghost reconstruction walk every edge of the ~150-vertex
interface polygon for every cell-centre and face query, every step.
(Also measured, refuting the parked consolidation: Level::new — the MG
hierarchy build — is 0.5-0.7% in BOTH phases; caching it would buy
nothing. The MG smoother at 34-40% is the honest remaining fluid cost.)

PolygonSdf (solvers/incompressible/polygon_sdf.rs): a binned edge
index whose query is BIT-IDENTICAL to polygon_signed_distance by
construction — per-edge distances use the same float ops, the ring
search provably visits a superset of the argmin (convex-projection
lower bound sqrt(d_out^2 + ((r-1)b)^2)), and parity XORs the same ray
tests over exactly the straddling edges (y-binned). Equality is
ASSERTED, not assumed: tests compare to_bits against the brute force
over ~40k adversarial points (flag-like walks, random polygons with
degenerate zero-length edges, horizontal-edge/vertex-y rays). Wired
into EmbeddedBody::polygon and the FSI harness's shared geometry
(rebuilt per set_geometry, ~microseconds for 150 edges).

Verification — the bar for a bit-exact change is digit identity, and
it holds: FSI2 and FSI3 committed defaults reproduce EVERY printed
digit of the banded-LU baseline logs (uy 3.7732±3.7920 / 6.0229±
25.2190 mm, conservation 8.26e-12 / 1.49e-12, rigid drags 121.4 /
426.9); rtx-cfd full suite 0 failures; rtx-fsi lib/piston/transfer/
FSI1 green. The study pins need no re-run: the trajectories are
unchanged by construction and confirmed by measurement.

Wall clock: FSI2 rigid 323 -> 167 s (1.93x), whole default 400 -> 225 s;
FSI3 rigid 420 -> 250 s (1.68x), whole default 539 -> 343 s. Cumulative
with the banded LU this session: FSI3 default 944 -> 343 s (2.75x),
FSI2 524 -> 225 s (2.33x).

Co-Authored-By: Claude Fable 5 <[email protected]>
Claude-Session: https://claude.ai/code/session_01X2GmJXeQ2njUecEKiJZ1G2

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded_body.rs

@@ -173,8 +173,11 @@ impl EmbeddedBody {
             .sum::<f64>()
             * 0.5;
         let ccw = signed_area > 0.0;
-        let sdf_vertices = vertices.clone();
+        // Indexed query, bit-identical to `polygon_signed_distance`
-        let mut body = Self::from_sdf(move |x, y, _| polygon_signed_distance(&sdf_vertices, x, y));
+        // (asserted in polygon_sdf's tests) — the SDF is the measured
+        // hot function of the embedded mask rebuild.
+        let sdf = super::polygon_sdf::PolygonSdf::new(vertices.clone());
+        let mut body = Self::from_sdf(move |x, y, _| sdf.signed_distance(x, y));
         let sampler_vertices = vertices;
         body.sampler = Some(Box::new(move |ds| {
             let n = sampler_vertices.len();

crates/specialized/rtx-cfd/src/solvers/incompressible/mod.rs

@@ -26,6 +26,7 @@ pub mod piso;
 pub mod piso_gpu;
 /// Five-point Poisson problems and the multigrid-preconditioned CG solver
 pub mod poisson;
+pub mod polygon_sdf;
 /// SIMPLE algorithm implementation
 pub mod simple;
 /// GPU-accelerated SIMPLE algorithm implementation
@@ -49,6 +50,7 @@ pub use piso::{PisoParameters, PisoResult, PisoSolver};
 #[cfg(feature = "cuda")]
 pub use piso_gpu::PisoGpuSolver;
 pub use poisson::{MultigridParameters, PoissonProblem, PoissonSolution, PoissonSolverKind};
+pub use polygon_sdf::PolygonSdf;
 pub use simple::{ConvectionScheme, SimpleParameters, SimpleResult, SimpleSolver};
 #[cfg(feature = "cuda")]
 pub use simple_gpu::SimpleGpuSolver;

crates/specialized/rtx-cfd/src/solvers/incompressible/polygon_sdf.rs

@@ -0,0 +1,335 @@
+// Copyright (c) 2024 RustyTorch++ Team
+// Licensed under the Apache License, Version 2.0
+
+//! An indexed polygon signed-distance query, bit-identical to
+//! [`polygon_signed_distance`].
+//!
+//! Measured motivation (2026-08-30 FSI3 profile): 51% of the fluid step
+//! is `polygon_signed_distance` — the embedded mask rebuild and its
+//! ghost reconstruction evaluate the SDF tens of thousands of times per
+//! step, each walking every edge of a ~150-vertex interface polygon.
+//! This index cuts each query to the handful of edges that can matter,
+//! while returning EXACTLY the brute-force f64:
+//!
+//! - **Distance**: per-edge squared distances are computed by the same
+//!   float ops as the brute force; the ring search visits a candidate
+//!   set that provably contains the minimizing edge, and `min` over a
+//!   superset containing the argmin equals `min` over all edges bit for
+//!   bit. The ring lower bound uses the convex-projection inequality:
+//!   for a query `q` clamped to `c` on the grid box and any point `z`
+//!   inside it, `|q−z|² ≥ |q−c|² + |c−z|²`, so a ring at Chebyshev
+//!   index `r` is at least `√(d_out² + ((r−1)·b)²)` away (`b` the
+//!   smaller bin dimension).
+//! - **Sign**: the even-odd ray test flips parity only for edges whose
+//!   y-interval straddles the query, so edges are binned by y-interval
+//!   and the query XORs over exactly the straddling candidates — the
+//!   same tests, the same parity.
+//!
+//! The equality is asserted, not assumed: the tests compare against
+//! [`polygon_signed_distance`] with `to_bits` over adversarial points.
+
+use super::embedded_body::polygon_signed_distance;
+
+/// Indexed signed distance to a closed polygon (negative inside, either
+/// winding). Build once per geometry; queries are `O(edges near the
+/// point)` instead of `O(all edges)`.
+pub struct PolygonSdf {
+    vertices: Vec<(f64, f64)>,
+    /// Grid over the polygon's padded bounding box.
+    x0: f64,
+    y0: f64,
+    bin_w: f64,
+    bin_h: f64,
+    nx: usize,
+    ny: usize,
+    /// Edge indices per bin (an edge appears in every bin its padded
+    /// bounding box overlaps).
+    bins: Vec<Vec<u32>>,
+    /// Edge indices per y-row of the SAME grid, for the ray-crossing
+    /// parity: an edge appears in every row its y-interval overlaps.
+    rows: Vec<Vec<u32>>,
+}
+
+impl PolygonSdf {
+    /// Index `vertices` (at least 3). Bin count scales with the edge
+    /// count so construction stays `O(edges)`.
+    #[must_use]
+    pub fn new(vertices: Vec<(f64, f64)>) -> Self {
+        assert!(vertices.len() >= 3, "a polygon needs at least 3 vertices");
+        let n = vertices.len();
+        let (mut min_x, mut min_y) = (f64::MAX, f64::MAX);
+        let (mut max_x, mut max_y) = (f64::MIN, f64::MIN);
+        for &(x, y) in &vertices {
+            min_x = min_x.min(x);
+            min_y = min_y.min(y);
+            max_x = max_x.max(x);
+            max_y = max_y.max(y);
+        }
+        // Degenerate extents still get a positive bin size.
+        let width = (max_x - min_x).max(1e-12);
+        let height = (max_y - min_y).max(1e-12);
+        // ~2 edges per bin on a perimeter polygon: bins ~ n along the
+        // longer side, aspect-scaled on the shorter.
+        let nx = ((n as f64).sqrt() * (width / height).sqrt().max(0.25))
+            .ceil()
+            .clamp(1.0, 256.0) as usize;
+        let ny = ((n as f64).sqrt() * (height / width).sqrt().max(0.25))
+            .ceil()
+            .clamp(1.0, 256.0) as usize;
+        let bin_w = width / nx as f64;
+        let bin_h = height / ny as f64;
+
+        let mut bins = vec![Vec::new(); nx * ny];
+        let mut rows = vec![Vec::new(); ny];
+        let clamp_i = |x: f64| (((x - min_x) / bin_w) as isize).clamp(0, nx as isize - 1) as usize;
+        let clamp_j = |y: f64| (((y - min_y) / bin_h) as isize).clamp(0, ny as isize - 1) as usize;
+        for k in 0..n {
+            let (ax, ay) = vertices[k];
+            let (bx, by) = vertices[(k + 1) % n];
+            let (i0, i1) = (clamp_i(ax.min(bx)), clamp_i(ax.max(bx)));
+            let (j0, j1) = (clamp_j(ay.min(by)), clamp_j(ay.max(by)));
+            for j in j0..=j1 {
+                for i in i0..=i1 {
+                    bins[j * nx + i].push(k as u32);
+                }
+                rows[j].push(k as u32);
+            }
+        }
+        Self {
+            vertices,
+            x0: min_x,
+            y0: min_y,
+            bin_w,
+            bin_h,
+            nx,
+            ny,
+            bins,
+            rows,
+        }
+    }
+
+    /// The indexed vertices.
+    #[must_use]
+    pub fn vertices(&self) -> &[(f64, f64)] {
+        &self.vertices
+    }
+
+    /// Squared distance from `(x, y)` to edge `k` — float-op for
+    /// float-op the brute force's per-edge computation.
+    #[inline]
+    fn edge_dist2(&self, k: u32, x: f64, y: f64) -> f64 {
+        let n = self.vertices.len();
+        let (ax, ay) = self.vertices[k as usize];
+        let (bx, by) = self.vertices[(k as usize + 1) % n];
+        let (ex, ey) = (bx - ax, by - ay);
+        let len2 = ex * ex + ey * ey;
+        let s = if len2 > 0.0 {
+            (((x - ax) * ex + (y - ay) * ey) / len2).clamp(0.0, 1.0)
+        } else {
+            0.0
+        };
+        let (qx, qy) = (ax + s * ex - x, ay + s * ey - y);
+        qx * qx + qy * qy
+    }
+
+    /// Signed distance, bit-identical to
+    /// `polygon_signed_distance(self.vertices(), x, y)`.
+    #[must_use]
+    pub fn signed_distance(&self, x: f64, y: f64) -> f64 {
+        // Sign: XOR the ray test over the y-row candidates. Any edge
+        // that straddles y lies in this row's list (its y-interval
+        // overlaps the row), so the parity is over exactly the edges
+        // the brute force flips on.
+        let j_row = (((y - self.y0) / self.bin_h) as isize).clamp(0, self.ny as isize - 1) as usize;
+        let mut inside = false;
+        // Edges whose y-interval leaves the grid entirely are impossible
+        // (the grid spans the polygon's bbox), but a query y outside the
+        // bbox straddles nothing — the clamped row still contains every
+        // straddling edge because there are none.
+        for &k in &self.rows[j_row] {
+            let n = self.vertices.len();
+            let (ax, ay) = self.vertices[k as usize];
+            let (bx, by) = self.vertices[(k as usize + 1) % n];
+            if (ay > y) != (by > y) {
+                let x_cross = ax + (y - ay) / (by - ay) * (bx - ax);
+                if x < x_cross {
+                    inside = !inside;
+                }
+            }
+        }
+
+        // Distance: ring search from the clamped bin.
+        let ci = (((x - self.x0) / self.bin_w) as isize).clamp(0, self.nx as isize - 1);
+        let cj = (((y - self.y0) / self.bin_h) as isize).clamp(0, self.ny as isize - 1);
+        // Distance from the query to the grid box (0 inside).
+        let cx = x.clamp(self.x0, self.x0 + self.bin_w * self.nx as f64);
+        let cy = y.clamp(self.y0, self.y0 + self.bin_h * self.ny as f64);
+        let d_out2 = (x - cx) * (x - cx) + (y - cy) * (y - cy);
+        let b = self.bin_w.min(self.bin_h);
+
+        let mut dist2 = f64::MAX;
+        let max_ring = self.nx.max(self.ny) as isize;
+        for r in 0..=max_ring {
+            // Every point of a ring-r bin is at least this far away
+            // (convex-projection inequality; see the module docs).
+            if r >= 2 {
+                let lb = (r - 1) as f64 * b;
+                if d_out2 + lb * lb > dist2 {
+                    break;
+                }
+            }
+            let (i_lo, i_hi) = (ci - r, ci + r);
+            let (j_lo, j_hi) = (cj - r, cj + r);
+            let mut visit = |i: isize, j: isize, dist2: &mut f64| {
+                if i < 0 || j < 0 || i >= self.nx as isize || j >= self.ny as isize {
+                    return;
+                }
+                for &k in &self.bins[j as usize * self.nx + i as usize] {
+                    let d2 = self.edge_dist2(k, x, y);
+                    if d2 < *dist2 {
+                        *dist2 = d2;
+                    }
+                }
+            };
+            if r == 0 {
+                visit(ci, cj, &mut dist2);
+            } else {
+                for i in i_lo..=i_hi {
+                    visit(i, j_lo, &mut dist2);
+                    visit(i, j_hi, &mut dist2);
+                }
+                for j in (j_lo + 1)..j_hi {
+                    visit(i_lo, j, &mut dist2);
+                    visit(i_hi, j, &mut dist2);
+                }
+            }
+        }
+
+        let dist = dist2.sqrt();
+        if inside { -dist } else { dist }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    /// A deterministic pseudo-random stream (no rand dependency).
+    struct Lcg(u64);
+    impl Lcg {
+        fn next_f64(&mut self, lo: f64, hi: f64) -> f64 {
+            self.0 = self
+                .0
+                .wrapping_mul(6364136223846793005)
+                .wrapping_add(1442695040888963407);
+            let u = (self.0 >> 11) as f64 / (1u64 << 53) as f64;
+            lo + u * (hi - lo)
+        }
+    }
+
+    fn assert_bit_identical(vertices: &[(f64, f64)], points: &[(f64, f64)]) {
+        let sdf = PolygonSdf::new(vertices.to_vec());
+        for &(x, y) in points {
+            let brute = polygon_signed_distance(vertices, x, y);
+            let indexed = sdf.signed_distance(x, y);
+            assert_eq!(
+                brute.to_bits(),
+                indexed.to_bits(),
+                "indexed {indexed:.17e} != brute {brute:.17e} at ({x}, {y})"
+            );
+        }
+    }
+
+    /// A flag-like polygon: a long thin rectangle sampled densely (the
+    /// FSI interface walk's shape), mildly deformed.
+    fn flag_polygon(n_per_side: usize, deflect: f64) -> Vec<(f64, f64)> {
+        let (x0, x1, y0, y1) = (0.25, 0.6, 0.19, 0.21);
+        let mut v = Vec::new();
+        for k in 0..n_per_side {
+            let s = k as f64 / n_per_side as f64;
+            let x = x0 + s * (x1 - x0);
+            v.push((x, y0 + deflect * s * s));
+        }
+        for k in 0..3 {
+            let s = k as f64 / 3.0;
+            v.push((x1, y0 + deflect + s * (y1 - y0)));
+        }
+        for k in 0..n_per_side {
+            let s = k as f64 / n_per_side as f64;
+            let x = x1 - s * (x1 - x0);
+            v.push((x, y1 + deflect * (1.0 - s) * (1.0 - s)));
+        }
+        v.push((x0, y1));
+        v
+    }
+
+    #[test]
+    fn bit_identical_on_flag_polygon() {
+        for &deflect in &[0.0, 0.05, -0.08] {
+            let vertices = flag_polygon(72, deflect);
+            let mut rng = Lcg(42);
+            let mut points = Vec::new();
+            // The whole domain, the near field, and exactly-on-feature
+            // points (vertices, edge midpoints, the bbox corners).
+            for _ in 0..2000 {
+                points.push((rng.next_f64(0.0, 2.5), rng.next_f64(0.0, 0.41)));
+            }
+            for _ in 0..2000 {
+                points.push((rng.next_f64(0.24, 0.62), rng.next_f64(0.15, 0.28)));
+            }
+            for k in 0..vertices.len() {
+                let (ax, ay) = vertices[k];
+                let (bx, by) = vertices[(k + 1) % vertices.len()];
+                points.push((ax, ay));
+                points.push((0.5 * (ax + bx), 0.5 * (ay + by)));
+            }
+            points.push((-3.0, -1.0));
+            points.push((10.0, 5.0));
+            assert_bit_identical(&vertices, &points);
+        }
+    }
+
+    #[test]
+    fn bit_identical_on_random_polygons() {
+        let mut rng = Lcg(7);
+        for poly in 0..20 {
+            let n = 3 + (poly % 9);
+            let mut vertices: Vec<(f64, f64)> = (0..n)
+                .map(|_| (rng.next_f64(-1.0, 1.0), rng.next_f64(-1.0, 1.0)))
+                .collect();
+            // Exercise degenerate zero-length edges too.
+            if poly % 4 == 0 {
+                let first = vertices[0];
+                vertices.insert(1, first);
+            }
+            let points: Vec<(f64, f64)> = (0..1500)
+                .map(|_| (rng.next_f64(-3.0, 3.0), rng.next_f64(-3.0, 3.0)))
+                .collect();
+            assert_bit_identical(&vertices, &points);
+        }
+    }
+
+    #[test]
+    fn bit_identical_on_horizontal_edge_rays() {
+        // Horizontal edges never straddle their own y (the strict/loose
+        // comparison pair `(ay > y) != (by > y)` is false when ay == by),
+        // and queries exactly AT a vertex y exercise the boundary of the
+        // straddle test. The parity must match the brute force on all of
+        // them.
+        let vertices = vec![
+            (0.0, 0.0),
+            (2.0, 0.0),
+            (2.0, 1.0),
+            (1.0, 1.0),
+            (1.0, 0.5),
+            (0.0, 0.5),
+        ];
+        let mut points = Vec::new();
+        for &y in &[0.0, 0.25, 0.5, 0.75, 1.0] {
+            for k in 0..40 {
+                points.push((-0.5 + 3.0 * k as f64 / 39.0, y));
+            }
+        }
+        assert_bit_identical(&vertices, &points);
+    }
+}

crates/specialized/rtx-fsi/tests/fsi2_harness/mod.rs

@@ -19,7 +19,7 @@ use nalgebra::Vector3;
 use rtx_cfd::CfdConfig;
 use rtx_cfd::solvers::incompressible::{
     AleBoundaries, ConvectionScheme, EmbeddedBody, EmbeddedParameters, EmbeddedPisoSolver,
-    FlowField, PoissonSolverKind, SideBoundary, polygon_interface_velocity,
+    FlowField, PoissonSolverKind, PolygonSdf, SideBoundary, polygon_interface_velocity,
     polygon_signed_distance,
 };
 use rtx_fea::assembly::dof_mapping::DofComponent;
@@ -334,7 +334,7 @@ pub struct Fsi2Harness {
     /// The deformable geometry AND its velocity, behind one lock: the
     /// fluid's per-step mask rebuild reads the polygon; the no-slip
     /// closure reads both.
-    pub shared: Arc<RwLock<(Vec<(f64, f64)>, Vec<(f64, f64)>)>>,
+    pub shared: Arc<RwLock<(PolygonSdf, Vec<(f64, f64)>)>>,
     pub spiked_total: Cell<usize>,
 }
 
@@ -377,8 +377,11 @@ impl Fsi2Harness {
             .expect("point A");
 
         let zero_d = vec![0.0; 2 * interface.wetted.len()];
+        // The polygon lives behind the lock as an INDEXED SDF
+        // (bit-identical query; the brute-force walk was measured at
+        // 51% of the fluid step, called for every mask cell and ghost).
         let shared = Arc::new(RwLock::new((
-            interface.polygon(&zero_d),
+            PolygonSdf::new(interface.polygon(&zero_d)),
             interface.walk_velocities(&zero_d),
         )));
         let sdf_shared = shared.clone();
@@ -418,14 +421,14 @@ impl Fsi2Harness {
         solver.set_moving_body(
             EmbeddedBody::from_sdf(move |x, y, _| {
                 let geometry = sdf_shared.read().unwrap();
-                circle_sdf(x, y).min(polygon_signed_distance(&geometry.0, x, y))
+                circle_sdf(x, y).min(geometry.0.signed_distance(x, y))
             })
             .with_surface_velocity(move |x, y, _| {
                 let geometry = vel_shared.read().unwrap();
-                if circle_sdf(x, y) <= polygon_signed_distance(&geometry.0, x, y) {
+                if circle_sdf(x, y) <= geometry.0.signed_distance(x, y) {
                     (0.0, 0.0)
                 } else {
-                    polygon_interface_velocity(&geometry.0, &geometry.1, x, y)
+                    polygon_interface_velocity(geometry.0.vertices(), &geometry.1, x, y)
                 }
             }),
         );
@@ -454,7 +457,7 @@ impl Fsi2Harness {
     /// Publish an interface geometry (+ velocity) to the fluid.
     pub fn set_geometry(&self, d: &[f64], ddot: &[f64]) {
         let mut geometry = self.shared.write().unwrap();
-        geometry.0 = self.interface.polygon(d);
+        geometry.0 = PolygonSdf::new(self.interface.polygon(d));
         geometry.1 = self.interface.walk_velocities(ddot);
     }
 
@@ -462,7 +465,7 @@ impl Fsi2Harness {
     pub fn measure_force(&self, solver: &EmbeddedPisoSolver, field: &FlowField) -> (f64, f64) {
         let mask = solver.mask().unwrap();
         let body = solver.body().unwrap();
-        let vertices = self.shared.read().unwrap().0.clone();
+        let vertices = self.shared.read().unwrap().0.vertices().to_vec();
         // Collect, then clamp, then integrate: the same 20x-median spike
         // clamp the coupling loads carry. Without it the REPORTED
         // drag/lift at large deformation are dominated by the rare wild