embedded3: study knobs — aperture_substeps (space-time sub-sampling), momentum_volume_cell_mean, RTX_E3_MERGE_FRACTION; all default off (refuted as the fresh-row residual's source)

Co-Authored-By: Claude Fable 5.1 <[email protected]>

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/cutwall.rs

@@ -228,8 +228,13 @@ impl Mask {
             old.and_then(|o| o.cut.as_ref())
                 .map_or(v, |oc| v.max(oc.vol[idx]))
         };
+        // `RTX_E3_MERGE_FRACTION` overrides the threshold (a study knob).
+        let threshold = std::env::var("RTX_E3_MERGE_FRACTION")
+            .ok()
+            .and_then(|v| v.parse().ok())
+            .unwrap_or(MERGE_FRACTION);
         let small: Vec<bool> = (0..n)
-            .map(|idx| self.cell_active(idx) && frac(idx) < MERGE_FRACTION)
+            .map(|idx| self.cell_active(idx) && frac(idx) < threshold)
             .collect();
         let mut master = vec![usize::MAX; n];
         for idx in (0..n).filter(|&i| small[i]) {
@@ -256,17 +261,37 @@ impl Mask {
     }
 
     /// Set the step-averaged apertures and the space-time classification
-    /// from the previous mask's geometry.
+    /// from the previous mask's geometry: the trapezoid `½(αⁿ + αⁿ⁺¹)`,
+    /// or with `inner` intermediate geometries the composite trapezoid
+    /// over the step (the exact time integral of the space-time cut cell,
+    /// arXiv 2512.23358, approached as the sub-sampling refines).
     pub fn set_step_apertures(&mut self, old: &Mask) {
+        self.set_step_apertures_with(old, &[]);
+    }
+
+    /// As [`Self::set_step_apertures`] with the apertures of the
+    /// intermediate geometries `inner` (in time order) inside the step.
+    pub fn set_step_apertures_with(&mut self, old: &Mask, inner: &[&CutGeometry]) {
         let (Some(cut), Some(old_cut)) = (self.cut.as_ref(), old.cut.as_ref()) else {
             return;
         };
-        let avg = |a: &[f64], b: &[f64]| -> Vec<f64> {
+        let n = inner.len() + 1;
-            a.iter().zip(b).map(|(x, y)| 0.5 * (x + y)).collect()
+        let w_end = 0.5 / n as f64;
+        let w_in = 1.0 / n as f64;
+        let avg = |pick: &dyn Fn(&CutGeometry) -> &[f64]| -> Vec<f64> {
+            let a = pick(cut);
+            let b = pick(old_cut);
+            let mut out: Vec<f64> = a.iter().zip(b).map(|(x, y)| w_end * (x + y)).collect();
+            for g in inner {
+                for (o, v) in out.iter_mut().zip(pick(g)) {
+                    *o += w_in * v;
+                }
+            }
+            out
         };
-        let au = avg(&cut.a_u, &old_cut.a_u);
+        let au = avg(&|g: &CutGeometry| &g.a_u);
-        let av = avg(&cut.a_v, &old_cut.a_v);
+        let av = avg(&|g: &CutGeometry| &g.a_v);
-        let aw = avg(&cut.a_w, &old_cut.a_w);
+        let aw = avg(&|g: &CutGeometry| &g.a_w);
         let open = |a: &[f64]| -> Vec<bool> { a.iter().map(|&x| x > 0.0).collect() };
         let active = self
             .cell_fluid

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/cut_predictor.rs

@@ -213,7 +213,13 @@ impl Solver {
         let a_w =
             (cv.wall[0] * cv.wall[0] + cv.wall[1] * cv.wall[1] + cv.wall[2] * cv.wall[2]).sqrt();
         let shear = mu * a_w / cv.distance;
-        let v_eff = cv.alpha.max(INERTIA_FLOOR) * h[c] * area[c];
+        let fraction = if self.params.momentum_volume_cell_mean {
+            let vol = |q: [i64; 3]| lat.cell(q).map_or(cv.alpha, |ci| mask.vol(ci));
+            0.5 * (vol(cell_minus) + vol(cell_plus))
+        } else {
+            cv.alpha
+        };
+        let v_eff = fraction.max(INERTIA_FLOOR) * h[c] * area[c];
         let inertia = rho * v_eff / dt;
         (inertia * u0 - conv + diff + pressure + source + shear * ub) / (inertia + shear)
     }

crates/specialized/rtx-cfd/src/solvers/incompressible/embedded3/step/mod.rs

@@ -12,6 +12,7 @@ mod projection;
 
 use super::Grid;
 use super::body::Body;
+use super::cut::CutGeometry;
 use super::field::Field;
 use super::poisson::{PcgCache, Problem, solve_pcg_cached};
 use super::wall::{FaceKind, Mask, WallScheme};
@@ -79,6 +80,16 @@ pub struct Parameters {
     /// (identical results; the far corners keep their sign). `None` =
     /// every corner every step.
     pub max_surface_speed: Option<f64>,
+    /// Intermediate geometries per step for the space-time apertures
+    /// (0 = the trapezoid of the two end points).
+    pub aperture_substeps: usize,
+    /// The cut predictor's momentum volume per unit `h A`: the face's own
+    /// aperture (the registered design, `false`) or the mean fluid
+    /// fraction of the two cells the face separates (`true`: the control
+    /// volume's actual fluid content — a fresh face then carries the
+    /// inertia of the fluid half of its volume instead of a tenth of a
+    /// cell against O(1) fluxes).
+    pub momentum_volume_cell_mean: bool,
 }
 
 impl Default for Parameters {
@@ -93,6 +104,8 @@ impl Default for Parameters {
             inner_stop_factor: 1e-2,
             wall_scheme: WallScheme::GhostBinary,
             max_surface_speed: None,
+            aperture_substeps: 0,
+            momentum_volume_cell_mean: false,
         }
     }
 }
@@ -502,7 +515,32 @@ impl Solver {
         let mut new_mask = self.build_mask(body, field.grid, t_new, dt);
         if let Some(old_mask) = &self.mask {
             fresh_cells = refill_fresh_cells(old_mask, &new_mask, field);
-            new_mask.set_step_apertures(old_mask);
+            let n_in = self.params.aperture_substeps;
+            if n_in == 0 {
+                new_mask.set_step_apertures(old_mask);
+            } else {
+                // The intermediate geometries, each within the band of the
+                // previous one.
+                let g = field.grid;
+                let h = g.dx.min(g.dy).min(g.dz);
+                let t_old = t_new - dt;
+                let mut inner: Vec<CutGeometry> = Vec::with_capacity(n_in);
+                for m in 1..=n_in {
+                    let tm = t_old + dt * m as f64 / (n_in + 1) as f64;
+                    let prev: Option<(&CutGeometry, f64, f64)> = match (
+                        self.params.max_surface_speed,
+                        inner.last().or(old_mask.cut()),
+                    ) {
+                        (Some(speed), Some(c)) => {
+                            Some((c, 3.0 * h, speed * dt / (n_in + 1) as f64))
+                        }
+                        _ => None,
+                    };
+                    inner.push(CutGeometry::build_from(body, g, tm, prev));
+                }
+                let refs: Vec<&CutGeometry> = inner.iter().collect();
+                new_mask.set_step_apertures_with(old_mask, &refs);
+            }
         }
         new_mask.impose_from(
             body,

crates/specialized/rtx-cfd/tests/embedded3_falsifier.rs

@@ -241,6 +241,14 @@ fn run(scheme: WallScheme, moving: bool, dt: f64, t_end: f64) -> Run {
                 z1: Side::Periodic,
                 ..Boundaries::default()
             },
+            max_surface_speed: Some(U_PEAK),
+            // `RTX_E3_SUBSTEPS`: intermediate geometries for the space-time apertures.
+            aperture_substeps: std::env::var("RTX_E3_SUBSTEPS")
+                .ok()
+                .and_then(|v| v.parse().ok())
+                .unwrap_or(0),
+            // `RTX_E3_CELL_MEAN=1`: the momentum volume from the two cells' fractions.
+            momentum_volume_cell_mean: std::env::var("RTX_E3_CELL_MEAN").is_ok_and(|v| v == "1"),
             ..Parameters::default()
         },
     );

crates/specialized/rtx-cfd/tests/embedded3_sphere/mod.rs

@@ -133,6 +133,7 @@ pub fn measure(n: usize, scheme: WallScheme, c: (f64, f64, f64)) -> Measurement
             tolerance: 1e-8,
             convection_scheme: ConvectionScheme::Upwind,
             wall_scheme: scheme,
+            momentum_volume_cell_mean: std::env::var("RTX_E3_CELL_MEAN").is_ok_and(|v| v == "1"),
             ..Parameters::default()
         },
     );