rtx-cfd embedded3 item 9: wall.rs (binary ghost mask: three face families, trilinear stencil with periodic-z wrap, z-weighted least-squares ghost fit, flux compatibility correction; slip walls allowed as touched sides) + loads.rs (surface-stress route with probes, control-volume route with full-span z faces skipped); the step wired (body/mask, predicates, anchor, ghost re-imposition). Gates HELD: CFD1 ny 41 nz 1 CV 15.6156 / surface 15.7126 both to 1e-6 of the 2D record; nz 4 periodic CV 1.1e-6 / surface 4.2e-4; sphere MMS order 0.89, div 1e-8, ghost correction 1.0e-4 → 2.0e-5, both load routes' errors falling (0.153 → 0.115 surface, 0.214 → 0.139 CV)

Co-Authored-By: Claude Fable 5.1 <[email protected]>
2026-09-17 15:14:40 -05:00
parent d4ffac9ac7
commit d337afa8f9
7 changed files with 1796 additions and 12 deletions
@@ -0,0 +1,406 @@
+//! The two load routes of the 2D solver in 3D. Route A, surface stress
+//! reconstruction: at each surface sample the pressure is linearly
+//! extrapolated to the wall from probes at `h` and `2h` along the normal;
+//! the wall-normal derivatives of the normal and two tangential velocity
+//! components from the same probes with the surface velocity at the wall
+//! (quadratic fit); the tangential derivatives of the normal velocity from
+//! the surface-velocity function; `σ·n = (−p + 2μ ∂ₙuₙ) n + μ Σₜ (∂ₙuₜ +
+//! ∂ₜuₙ) t`. Route B, a momentum balance over a box of whole cells
+//! enclosing the body — it reads no near-wall value; the two are unrelated
+//! readings of one solution.
+
+use super::body::Body;
+use super::field::Field;
+use super::wall::{FaceKind, Mask, linear_fit, stencil_nodes, z_planes};
+
+#[derive(Debug, Clone, Copy, PartialEq)]
+pub struct SurfaceForce {
+    pub f: [f64; 3],
+    pub samples: usize,
+    pub skipped: usize,
+}
+
+fn cross(a: [f64; 3], b: [f64; 3]) -> [f64; 3] {
+    [
+        a[1] * b[2] - a[2] * b[1],
+        a[2] * b[0] - a[0] * b[2],
+        a[0] * b[1] - a[1] * b[0],
+    ]
+}
+
+fn dot(a: [f64; 3], b: [f64; 3]) -> f64 {
+    a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
+}
+
+/// Two unit tangents orthonormal to `n`.
+fn tangents(n: [f64; 3]) -> ([f64; 3], [f64; 3]) {
+    let a = if n[0].abs() < 0.9 {
+        [1.0, 0.0, 0.0]
+    } else {
+        [0.0, 1.0, 0.0]
+    };
+    let t1 = cross(n, a);
+    let l = dot(t1, t1).sqrt();
+    let t1 = [t1[0] / l, t1[1] / l, t1[2] / l];
+    (t1, cross(n, t1))
+}
+
+impl Mask {
+    /// Pressure at a point from the cell centres: trilinear when the
+    /// surrounding cells are fluid, else the least-squares fit through the
+    /// fluid ones; `None` if degenerate.
+    pub fn pressure_at(&self, p: &[f64], x: f64, y: f64, z: f64) -> Option<f64> {
+        let g = self.grid();
+        let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
+        let (gx, gy, gz) = (x / dx - 0.5, y / dy - 0.5, z / dz - 0.5);
+        let i0 = gx.floor().clamp(0.0, (nx - 2) as f64) as usize;
+        let j0 = gy.floor().clamp(0.0, (ny - 2) as f64) as usize;
+        let fx = (gx - i0 as f64).clamp(0.0, 1.0);
+        let fy = (gy - j0 as f64).clamp(0.0, 1.0);
+        let (k0, k1, fz) = z_planes(gz, nz, self.periodic_z());
+        // A query on a single weighted plane lives on it (the fit has no
+        // z variation to determine).
+        let z = if nz == 1 || fz == 0.0 {
+            (k0 as f64 + 0.5) * dz
+        } else if fz == 1.0 {
+            (k1 as f64 + 0.5) * dz
+        } else {
+            z
+        };
+        let dirs = |m: usize, f: f64| {
+            if m <= 1 {
+                vec![(0usize, 1.0)]
+            } else {
+                vec![(0, 1.0 - f), (1, f)]
+            }
+        };
+        let kz = |dk: usize| if dk == 0 { k0 } else { k1 };
+        let mut nodes = Vec::with_capacity(8);
+        for (dk, wk) in dirs(nz, fz) {
+            for (dj, wj) in dirs(ny, fy) {
+                for (di, wi) in dirs(nx, fx) {
+                    nodes.push((g.cell(kz(dk), j0 + dj, i0 + di), wi * wj * wk));
+                }
+            }
+        }
+        if nodes.iter().all(|&(idx, _)| self.is_fluid_cell(idx)) {
+            return Some(nodes.iter().map(|&(idx, w)| w * p[idx]).sum());
+        }
+        // Weighted by the z-direction weight (a z-invariant field then fits
+        // as the 2D four-cell fit); the z weight of node `(dk, dj, di)` is
+        // `wk`, recovered from the trilinear product.
+        let zw = |wk: f64| wk;
+        let mut pts: Vec<(f64, f64, f64, f64, f64)> = Vec::new();
+        let mut it = nodes.iter();
+        for (dk, wk) in dirs(nz, fz) {
+            for _ in 0..dirs(ny, fy).len() * dirs(nx, fx).len() {
+                let &(idx, _) = it.next().expect("node");
+                let _ = dk;
+                if self.is_fluid_cell(idx) {
+                    let (k, j, i) = g.kji(idx);
+                    pts.push((
+                        (i as f64 + 0.5) * dx,
+                        (j as f64 + 0.5) * dy,
+                        (k as f64 + 0.5) * dz,
+                        p[idx],
+                        zw(wk),
+                    ));
+                }
+            }
+        }
+        linear_fit(&pts, (x, y, z))
+    }
+
+    /// Velocity at a point: trilinear over a component's nodes when all are
+    /// fluid faces, else the fit through the fluid ones and the point's own
+    /// boundary intercept with its surface velocity.
+    pub fn velocity_at(
+        &self,
+        body: &Body,
+        f: &Field,
+        x: f64,
+        y: f64,
+        z: f64,
+        t: f64,
+    ) -> Option<[f64; 3]> {
+        let g = self.grid();
+        let eps = 1e-6 * g.dx.min(g.dy).min(g.dz);
+        let s = body.phi(x, y, z, t);
+        let (n1, n2, n3) = body.normal(x, y, z, t, eps);
+        let foot = (x - s * n1, y - s * n2, z - s * n3);
+        let vf = body.surface_velocity(foot.0, foot.1, foot.2, t);
+        let vel_foot = [vf.0, vf.1, vf.2];
+        let mut out = [0.0; 3];
+        for c in 0..3 {
+            // A query on a single weighted plane of this component's nodes
+            // lives on it (the fit has no z variation to determine).
+            let (planes, offset) = if c == 2 {
+                (if self.periodic_z() { g.nz } else { g.nz + 1 }, 0.0)
+            } else {
+                (g.nz, 0.5)
+            };
+            let gz = z / g.dz - offset;
+            let (k0, k1, fz) = z_planes(gz, planes, self.periodic_z());
+            let zq = if planes <= 1 || fz == 0.0 {
+                (k0 as f64 + offset) * g.dz
+            } else if fz == 1.0 {
+                (k1 as f64 + offset) * g.dz
+            } else {
+                z
+            };
+            let foot_c = (foot.0, foot.1, if zq == z { foot.2 } else { zq });
+            let nodes = stencil_nodes((x, y, zq), c, g, self.periodic_z(), |_| None);
+            let values: &[f64] = [&f.u, &f.v, &f.w][c];
+            let fluid = |idx: usize| match c {
+                0 => self.u_kind(idx) == FaceKind::Fluid,
+                1 => self.v_kind(idx) == FaceKind::Fluid,
+                _ => self.w_kind(idx) == FaceKind::Fluid,
+            };
+            if nodes.iter().all(|n| fluid(n.idx)) {
+                out[c] = nodes.iter().map(|n| n.weight * values[n.idx]).sum();
+            } else {
+                let mut pts: Vec<(f64, f64, f64, f64, f64)> = nodes
+                    .iter()
+                    .filter(|n| fluid(n.idx))
+                    .map(|n| (n.x, n.y, n.z, values[n.idx], n.zw))
+                    .collect();
+                pts.push((foot_c.0, foot_c.1, foot_c.2, vel_foot[c], 1.0));
+                out[c] = linear_fit(&pts, (x, y, zq))?;
+            }
+        }
+        Some(out)
+    }
+
+    /// The reconstructed traction at one surface point with outward normal `n`.
+    pub fn traction_at(
+        &self,
+        body: &Body,
+        f: &Field,
+        mu: f64,
+        t: f64,
+        x: [f64; 3],
+        n: [f64; 3],
+    ) -> Option<[f64; 3]> {
+        let g = self.grid();
+        let h = g.dx.min(g.dy).min(g.dz);
+        let (d1, d2) = (h, 2.0 * h);
+        let at = |d: f64| [x[0] + d * n[0], x[1] + d * n[1], x[2] + d * n[2]];
+        let (x1, x2) = (at(d1), at(d2));
+        let p1 = self.pressure_at(&f.p, x1[0], x1[1], x1[2])?;
+        let p2 = self.pressure_at(&f.p, x2[0], x2[1], x2[2])?;
+        let u1 = self.velocity_at(body, f, x1[0], x1[1], x1[2], t)?;
+        let u2 = self.velocity_at(body, f, x2[0], x2[1], x2[2], t)?;
+        let p_wall = p1 + (p1 - p2) * d1 / (d2 - d1);
+        let (t1, t2) = tangents(n);
+        let us = body.surface_velocity(x[0], x[1], x[2], t);
+        let us = [us.0, us.1, us.2];
+        let wall_gradient =
+            |f1: f64, f2: f64| (f1 * d2 * d2 - f2 * d1 * d1) / (d1 * d2 * (d2 - d1));
+        let dn =
+            |dir: [f64; 3]| wall_gradient(dot(u1, dir) - dot(us, dir), dot(u2, dir) - dot(us, dir));
+        let eps = 1e-6 * h;
+        let dt_un = |dir: [f64; 3]| {
+            let p = body.surface_velocity(
+                x[0] + eps * dir[0],
+                x[1] + eps * dir[1],
+                x[2] + eps * dir[2],
+                t,
+            );
+            let m = body.surface_velocity(
+                x[0] - eps * dir[0],
+                x[1] - eps * dir[1],
+                x[2] - eps * dir[2],
+                t,
+            );
+            ((p.0 - m.0) * n[0] + (p.1 - m.1) * n[1] + (p.2 - m.2) * n[2]) / (2.0 * eps)
+        };
+        let traction_n = -p_wall + 2.0 * mu * dn(n);
+        let traction_t1 = mu * (dn(t1) + dt_un(t1));
+        let traction_t2 = mu * (dn(t2) + dt_un(t2));
+        Some([
+            traction_n * n[0] + traction_t1 * t1[0] + traction_t2 * t2[0],
+            traction_n * n[1] + traction_t1 * t1[1] + traction_t2 * t2[1],
+            traction_n * n[2] + traction_t1 * t1[2] + traction_t2 * t2[2],
+        ])
+    }
+
+    /// Route A: the surface integral of the reconstructed traction over the
+    /// body's samples at spacing `ds`.
+    pub fn surface_force(&self, body: &Body, f: &Field, mu: f64, t: f64, ds: f64) -> SurfaceForce {
+        let samples = body.surface_samples(ds);
+        let mut force = [0.0; 3];
+        let mut skipped = 0;
+        for s in &samples {
+            match self.traction_at(body, f, mu, t, [s.x, s.y, s.z], [s.nx, s.ny, s.nz]) {
+                Some(tr) => {
+                    for c in 0..3 {
+                        force[c] += tr[c] * s.area;
+                    }
+                }
+                None => skipped += 1,
+            }
+        }
+        SurfaceForce {
+            f: force,
+            samples: samples.len(),
+            skipped,
+        }
+    }
+
+    /// Route B: the momentum balance over the box of whole cells
+    /// `[i0, i1) × [j0, j1) × [k0, k1)` (in the fluid on its boundary):
+    /// `F = Σ_outer (σ·n − ρ u (u·n)) A − d/dt ∫ ρ u dV + ∫ f dV`.
+    #[allow(clippy::too_many_arguments)]
+    pub fn control_volume_force(
+        &self,
+        f: &Field,
+        dt: f64,
+        rho: f64,
+        mu: f64,
+        source: Option<&dyn Fn(f64, f64, f64) -> (f64, f64, f64)>,
+        (i0, i1, j0, j1, k0, k1): (usize, usize, usize, usize, usize, usize),
+    ) -> [f64; 3] {
+        let g = self.grid();
+        let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
+        let (u, v, w, p) = (&f.u, &f.v, &f.w, &f.p);
+        let mut force = [0.0; 3];
+        let uc =
+            |k: usize, j: usize, i: usize| 0.5 * (u[g.uface(k, j, i)] + u[g.uface(k, j, i + 1)]);
+        let vc =
+            |k: usize, j: usize, i: usize| 0.5 * (v[g.vface(k, j, i)] + v[g.vface(k, j + 1, i)]);
+        let wc =
+            |k: usize, j: usize, i: usize| 0.5 * (w[g.wface(k, j, i)] + w[g.wface(k + 1, j, i)]);
+        // Central inside, one-sided at the domain edge.
+        let dd = |prev: Option<f64>, here: f64, next: Option<f64>, h: f64| match (prev, next) {
+            (Some(a), Some(b)) => (b - a) / (2.0 * h),
+            (None, Some(b)) => (b - here) / h,
+            (Some(a), None) => (here - a) / h,
+            (None, None) => 0.0,
+        };
+        // x faces: u lives there.
+        for k in k0..k1 {
+            for j in j0..j1 {
+                for (i, sign) in [(i1, 1.0), (i0, -1.0)] {
+                    let un = u[g.uface(k, j, i)];
+                    let p_f = 0.5 * (p[g.cell(k, j, i - 1)] + p[g.cell(k, j, i)]);
+                    let dudx = (u[g.uface(k, j, i + 1)] - u[g.uface(k, j, i - 1)]) / (2.0 * dx);
+                    let dvdx = (vc(k, j, i) - vc(k, j, i - 1)) / dx;
+                    let dwdx = (wc(k, j, i) - wc(k, j, i - 1)) / dx;
+                    let dudy = dd(
+                        (j > 0).then(|| u[g.uface(k, j - 1, i)]),
+                        un,
+                        (j + 1 < ny).then(|| u[g.uface(k, j + 1, i)]),
+                        dy,
+                    );
+                    let dudz = dd(
+                        (k > 0).then(|| u[g.uface(k - 1, j, i)]),
+                        un,
+                        (k + 1 < nz).then(|| u[g.uface(k + 1, j, i)]),
+                        dz,
+                    );
+                    let v_f = 0.5 * (vc(k, j, i - 1) + vc(k, j, i));
+                    let w_f = 0.5 * (wc(k, j, i - 1) + wc(k, j, i));
+                    let a = dy * dz;
+                    force[0] += sign * ((-p_f + 2.0 * mu * dudx) - rho * un * un) * a;
+                    force[1] += sign * (mu * (dudy + dvdx) - rho * v_f * un) * a;
+                    force[2] += sign * (mu * (dudz + dwdx) - rho * w_f * un) * a;
+                }
+            }
+        }
+        // y faces: v lives there.
+        for k in k0..k1 {
+            for i in i0..i1 {
+                for (j, sign) in [(j1, 1.0), (j0, -1.0)] {
+                    let vn = v[g.vface(k, j, i)];
+                    let p_f = 0.5 * (p[g.cell(k, j - 1, i)] + p[g.cell(k, j, i)]);
+                    let dvdy = (v[g.vface(k, j + 1, i)] - v[g.vface(k, j - 1, i)]) / (2.0 * dy);
+                    let dudy = (uc(k, j, i) - uc(k, j - 1, i)) / dy;
+                    let dwdy = (wc(k, j, i) - wc(k, j - 1, i)) / dy;
+                    let dvdx = dd(
+                        (i > 0).then(|| v[g.vface(k, j, i - 1)]),
+                        vn,
+                        (i + 1 < nx).then(|| v[g.vface(k, j, i + 1)]),
+                        dx,
+                    );
+                    let dvdz = dd(
+                        (k > 0).then(|| v[g.vface(k - 1, j, i)]),
+                        vn,
+                        (k + 1 < nz).then(|| v[g.vface(k + 1, j, i)]),
+                        dz,
+                    );
+                    let u_f = 0.5 * (uc(k, j - 1, i) + uc(k, j, i));
+                    let w_f = 0.5 * (wc(k, j - 1, i) + wc(k, j, i));
+                    let a = dx * dz;
+                    force[0] += sign * (mu * (dudy + dvdx) - rho * u_f * vn) * a;
+                    force[1] += sign * ((-p_f + 2.0 * mu * dvdy) - rho * vn * vn) * a;
+                    force[2] += sign * (mu * (dvdz + dwdy) - rho * w_f * vn) * a;
+                }
+            }
+        }
+        // z faces: w lives there. A box spanning the whole z range has its
+        // z faces on the domain's z sides: no momentum flux and no shear on
+        // a slip wall, and the two faces cancel on a periodic pair — skipped.
+        let full_span = k0 == 0 && k1 == nz;
+        for j in (j0..j1).filter(|_| !full_span) {
+            for i in i0..i1 {
+                for (k, sign) in [(k1, 1.0), (k0, -1.0)] {
+                    let wn = w[g.wface(k, j, i)];
+                    let p_f = 0.5 * (p[g.cell(k - 1, j, i)] + p[g.cell(k, j, i)]);
+                    let dwdz = (w[g.wface(k + 1, j, i)] - w[g.wface(k - 1, j, i)]) / (2.0 * dz);
+                    let dudz = (uc(k, j, i) - uc(k - 1, j, i)) / dz;
+                    let dvdz = (vc(k, j, i) - vc(k - 1, j, i)) / dz;
+                    let dwdx = dd(
+                        (i > 0).then(|| w[g.wface(k, j, i - 1)]),
+                        wn,
+                        (i + 1 < nx).then(|| w[g.wface(k, j, i + 1)]),
+                        dx,
+                    );
+                    let dwdy = dd(
+                        (j > 0).then(|| w[g.wface(k, j - 1, i)]),
+                        wn,
+                        (j + 1 < ny).then(|| w[g.wface(k, j + 1, i)]),
+                        dy,
+                    );
+                    let u_f = 0.5 * (uc(k - 1, j, i) + uc(k, j, i));
+                    let v_f = 0.5 * (vc(k - 1, j, i) + vc(k, j, i));
+                    let a = dx * dy;
+                    force[0] += sign * (mu * (dudz + dwdx) - rho * u_f * wn) * a;
+                    force[1] += sign * (mu * (dvdz + dwdy) - rho * v_f * wn) * a;
+                    force[2] += sign * ((-p_f + 2.0 * mu * dwdz) - rho * wn * wn) * a;
+                }
+            }
+        }
+        // Unsteady term and source over the fluid cells of the box.
+        let dv = dx * dy * dz;
+        for k in k0..k1 {
+            for j in j0..j1 {
+                for i in i0..i1 {
+                    let idx = g.cell(k, j, i);
+                    if !self.is_fluid_cell(idx) {
+                        continue;
+                    }
+                    let (fu0, fu1) = (g.uface(k, j, i), g.uface(k, j, i + 1));
+                    let (fv0, fv1) = (g.vface(k, j, i), g.vface(k, j + 1, i));
+                    let (fw0, fw1) = (g.wface(k, j, i), g.wface(k + 1, j, i));
+                    let du = 0.5 * ((u[fu0] - f.u_old[fu0]) + (u[fu1] - f.u_old[fu1]));
+                    let dvv = 0.5 * ((v[fv0] - f.v_old[fv0]) + (v[fv1] - f.v_old[fv1]));
+                    let dw = 0.5 * ((w[fw0] - f.w_old[fw0]) + (w[fw1] - f.w_old[fw1]));
+                    force[0] -= rho * du / dt * dv;
+                    force[1] -= rho * dvv / dt * dv;
+                    force[2] -= rho * dw / dt * dv;
+                    if let Some(s) = source {
+                        let (sx, sy, sz) = s(
+                            (i as f64 + 0.5) * dx,
+                            (j as f64 + 0.5) * dy,
+                            (k as f64 + 0.5) * dz,
+                        );
+                        force[0] += sx * dv;
+                        force[1] += sy * dv;
+                        force[2] += sz * dv;
+                    }
+                }
+            }
+        }
+        force
+    }
+}
@@ -10,11 +10,15 @@ pub mod body;
 pub mod cut;
 pub mod field;
 pub mod grid;
+pub mod loads;
 pub mod poisson;
 pub mod step;
+pub mod wall;

 pub use body::{Body, SurfaceSample};
 pub use cut::CutGeometry;
 pub use field::Field;
 pub use grid::Grid;
+pub use loads::SurfaceForce;
 pub use step::{Boundaries, Fluid, Parameters, Side, Solver, StepResult};
+pub use wall::{FaceKind, Mask, WallScheme};
@@ -10,8 +10,10 @@ mod predictor;
 mod projection;

 use super::Grid;
+use super::body::Body;
 use super::field::Field;
 use super::poisson::{PcgCache, Problem, solve_pcg_cached};
+use super::wall::{FaceKind, Mask, WallScheme};
 use crate::solvers::incompressible::poisson::{
    MgPrecision, MgSmoother, MultigridParameters, PoissonSolution,
 };
@@ -40,7 +42,7 @@ pub struct Boundaries {
 }

 impl Boundaries {
-    fn any_outlet(self) -> bool {
+    pub(crate) fn any_outlet(self) -> bool {
        [self.x0, self.x1, self.y0, self.y1, self.z0, self.z1].contains(&Side::PressureOutlet)
    }

@@ -69,6 +71,8 @@ pub struct Parameters {
    pub convection_scheme: ConvectionScheme,
    /// Relative part of the pressure solve's inner stop (the 2D 1e-2).
    pub inner_stop_factor: f64,
+    /// The wall treatment of an embedded body.
+    pub wall_scheme: WallScheme,
 }

 impl Default for Parameters {
@@ -81,6 +85,7 @@ impl Default for Parameters {
            poisson_precision: MgPrecision::F64,
            convection_scheme: ConvectionScheme::Upwind,
            inner_stop_factor: 1e-2,
+            wall_scheme: WallScheme::GhostBinary,
        }
    }
 }
@@ -102,6 +107,9 @@ pub struct Solver {
    pub params: Parameters,
    pub(super) momentum_source: Option<Vec3Fn>,
    boundary_velocity: Option<Vec3Fn>,
+    body: Option<Body>,
+    mask: Option<Mask>,
+    last_ghost_correction: f64,
    pcg_cache: PcgCache,
    time: f64,
    initialized: bool,
@@ -126,6 +134,9 @@ impl Solver {
            params,
            momentum_source: None,
            boundary_velocity: None,
+            body: None,
+            mask: None,
+            last_ghost_correction: 0.0,
            pcg_cache: PcgCache::default(),
            time: 0.0,
            initialized: false,
@@ -147,6 +158,28 @@ impl Solver {
        self.boundary_velocity = Some(Box::new(f));
    }

+    /// A static embedded body (the mask is built at initialisation).
+    pub fn set_body(&mut self, body: Body) {
+        self.body = Some(body);
+        self.mask = None;
+    }
+
+    #[must_use]
+    pub fn body(&self) -> Option<&Body> {
+        self.body.as_ref()
+    }
+
+    #[must_use]
+    pub fn mask(&self) -> Option<&Mask> {
+        self.mask.as_ref()
+    }
+
+    /// The last step's ghost compatibility correction.
+    #[must_use]
+    pub fn ghost_correction(&self) -> f64 {
+        self.last_ghost_correction
+    }
+
    #[must_use]
    pub fn time(&self) -> f64 {
        self.time
@@ -167,22 +200,30 @@ impl Solver {
            .map_or((0.0, 0.0, 0.0), |f| f(x, y, z, t))
    }

-    // The fluid predicates: everything is fluid until the wall arrives.
+    // The fluid predicates (everything is fluid without a body).
    #[inline]
-    fn u_is_fluid(&self, _k: usize, _j: usize, _i: usize) -> bool {
-        true
+    pub(super) fn u_is_fluid(&self, k: usize, j: usize, i: usize) -> bool {
+        self.mask
+            .as_ref()
+            .is_none_or(|m| m.u_kind(m.grid().uface(k, j, i)) == FaceKind::Fluid)
    }
    #[inline]
-    fn v_is_fluid(&self, _k: usize, _j: usize, _i: usize) -> bool {
-        true
+    pub(super) fn v_is_fluid(&self, k: usize, j: usize, i: usize) -> bool {
+        self.mask
+            .as_ref()
+            .is_none_or(|m| m.v_kind(m.grid().vface(k, j, i)) == FaceKind::Fluid)
    }
    #[inline]
-    fn w_is_fluid(&self, _k: usize, _j: usize, _i: usize) -> bool {
-        true
+    pub(super) fn w_is_fluid(&self, k: usize, j: usize, i: usize) -> bool {
+        self.mask
+            .as_ref()
+            .is_none_or(|m| m.w_kind(m.grid().wface(k, j, i)) == FaceKind::Fluid)
    }
    #[inline]
-    fn cell_is_fluid(&self, _k: usize, _j: usize, _i: usize) -> bool {
-        true
+    pub(super) fn cell_is_fluid(&self, k: usize, j: usize, i: usize) -> bool {
+        self.mask
+            .as_ref()
+            .is_none_or(|m| m.is_fluid_cell(m.grid().cell(k, j, i)))
    }

    pub(super) fn upwind(face_velocity: f64, upstream: f64, downstream: f64) -> f64 {
@@ -318,10 +359,27 @@ impl Solver {
        field.copy_to_starred();
    }

-    /// Stamp the `t = time` boundary data (lazily called by the first step).
+    /// Build the mask (with a body) and stamp the `t = time` boundary data
+    /// and ghost values (lazily called by the first step).
    pub fn initialize(&mut self, field: &mut Field) {
        let t = self.time;
+        if let Some(body) = &self.body {
+            if self.mask.is_none() {
+                assert_eq!(
+                    self.params.wall_scheme,
+                    WallScheme::GhostBinary,
+                    "item 10 brings CutCell"
+                );
+                self.mask = Some(
+                    Mask::build(body, field.grid, t, self.params.boundaries)
+                        .expect("embedded mask"),
+                );
+            }
+        }
        self.apply_boundary_normals(field, t);
+        if let (Some(body), Some(mask)) = (&self.body, &self.mask) {
+            mask.impose(body, &mut field.u, &mut field.v, &mut field.w, t);
+        }
        self.initialized = true;
    }

@@ -354,6 +412,11 @@ impl Solver {
            }
            field.copy_to_starred();
        }
+        // Ghost faces follow the corrected field (the next step's stencil data).
+        if let (Some(body), Some(mask)) = (&self.body, &self.mask) {
+            self.last_ghost_correction =
+                mask.impose(body, &mut field.u, &mut field.v, &mut field.w, t_new);
+        }
        self.time = t_new;
        StepResult {
            converged: final_residual < self.params.tolerance,
@@ -93,7 +93,11 @@ impl Solver {
    /// The anchor cell of a pure-Neumann projection (the first fluid cell,
    /// the 2D `(1, 1)` at `k = 0`), or `None` with an outlet.
    pub(crate) fn anchor_cell(&self, g: Grid) -> Option<usize> {
-        (!self.params.boundaries.any_outlet()).then_some(g.cell(0, 1, 1))
+        (!self.params.boundaries.any_outlet()).then(|| {
+            self.mask
+                .as_ref()
+                .map_or(g.cell(0, 1, 1), super::super::wall::Mask::anchor)
+        })
    }

    /// The inner stop of a projection from the source scale (the 2D rule).
@@ -0,0 +1,655 @@
+//! The binary ghost wall — the 2D `EmbeddedMask` on the 3D grid. Cells are
+//! fluid where φ(centre) > 0; an interior face between two cells that are
+//! not both fluid is a ghost (within 1.5 h of the surface) or solid; a
+//! ghost face is prescribed the least-squares linear reconstruction
+//! through the fluid nodes of the trilinear stencil around the probe (one
+//! cell beyond the face's mirror image along the normal) plus the foot of
+//! the normal with its surface velocity — exact for linear fields — with
+//! the profile along the normal as the fallback; the ghost faces bounding
+//! fluid cells share a flux compatibility correction.
+
+use super::Grid;
+use super::body::Body;
+use super::step::{Boundaries, Side};
+
+#[derive(Debug, Clone, Copy, PartialEq, Eq)]
+pub enum FaceKind {
+    Fluid,
+    Ghost,
+    Solid,
+}
+
+/// The wall treatment of an embedded body.
+#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)]
+pub enum WallScheme {
+    /// The 2D binary mask ported (this module).
+    #[default]
+    GhostBinary,
+    /// The apertured cut-cell wall (item 10).
+    CutCell,
+}
+
+#[derive(Debug, Clone, Copy)]
+pub(crate) struct StencilNode {
+    pub(crate) idx: usize,
+    pub(crate) x: f64,
+    pub(crate) y: f64,
+    pub(crate) z: f64,
+    /// The trilinear weight (the fallback profile's interpolation).
+    pub(crate) weight: f64,
+    /// The z-direction weight alone: the least-squares weight of the node,
+    /// so that a z-invariant field fits exactly as the 2D four-node fit
+    /// (the two planes share the in-plane nodes' unit weight).
+    pub(crate) zw: f64,
+    pub(crate) fallback: Option<f64>,
+}
+
+#[derive(Debug, Clone)]
+struct Ghost {
+    idx: usize,
+    x: f64,
+    y: f64,
+    z: f64,
+    foot: (f64, f64, f64),
+    u_surface: f64,
+    s_face: f64,
+    s_probe: f64,
+    nodes: Vec<StencilNode>,
+    /// Outward-from-fluid sign for the compatibility correction (0 when no
+    /// fluid cell is adjacent).
+    flux_sign: f64,
+}
+
+#[derive(Clone)]
+pub struct Mask {
+    grid: Grid,
+    periodic_z: bool,
+    cell_fluid: Vec<bool>,
+    u_kind: Vec<FaceKind>,
+    v_kind: Vec<FaceKind>,
+    w_kind: Vec<FaceKind>,
+    u_ghosts: Vec<Ghost>,
+    v_ghosts: Vec<Ghost>,
+    w_ghosts: Vec<Ghost>,
+    anchor: usize,
+    fluid_cells: usize,
+}
+
+/// The z lattice position of a query: the lower plane index, the upper
+/// plane index and the weight of the upper plane. Periodic z wraps; a wall
+/// clamps (and a query outside the lattice takes the nearest plane).
+pub(crate) fn z_planes(gz: f64, planes: usize, periodic: bool) -> (usize, usize, f64) {
+    if planes <= 1 {
+        return (0, 0, 0.0);
+    }
+    if periodic {
+        let n = planes as f64;
+        let w = gz.rem_euclid(n);
+        let k0 = w.floor() as usize % planes;
+        ((k0) % planes, (k0 + 1) % planes, w - w.floor())
+    } else {
+        let k0 = gz.floor().clamp(0.0, (planes - 2) as f64) as usize;
+        (k0, k0 + 1, (gz - k0 as f64).clamp(0.0, 1.0))
+    }
+}
+
+/// The nodes of velocity component `c` (0 u, 1 v, 2 w) around `point`
+/// with trilinear weights (a single plane of nodes in a direction
+/// collapses to one node), clamped in x and y, wrapped or clamped in z;
+/// `fallback(idx)` supplies the value of a node that is not a fluid face.
+pub(crate) fn stencil_nodes(
+    point: (f64, f64, f64),
+    c: usize,
+    g: Grid,
+    periodic_z: bool,
+    fallback: impl Fn(usize) -> Option<f64>,
+) -> Vec<StencilNode> {
+    let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
+    // w faces on a periodic grid: the plane k = nz is the plane 0.
+    let (gx, gy, gz, max_i, max_j, planes_k) = match c {
+        0 => (
+            point.0 / dx,
+            point.1 / dy - 0.5,
+            point.2 / dz - 0.5,
+            nx,
+            ny - 1,
+            nz,
+        ),
+        1 => (
+            point.0 / dx - 0.5,
+            point.1 / dy,
+            point.2 / dz - 0.5,
+            nx - 1,
+            ny,
+            nz,
+        ),
+        _ => (
+            point.0 / dx - 0.5,
+            point.1 / dy - 0.5,
+            point.2 / dz,
+            nx - 1,
+            ny - 1,
+            if periodic_z { nz } else { nz + 1 },
+        ),
+    };
+    let i0 = gx.floor().clamp(0.0, (max_i.max(1) - 1) as f64) as usize;
+    let j0 = gy.floor().clamp(0.0, (max_j.max(1) - 1) as f64) as usize;
+    let fx = (gx - i0 as f64).clamp(0.0, 1.0);
+    let fy = (gy - j0 as f64).clamp(0.0, 1.0);
+    let (k0, k1, fz) = z_planes(gz, planes_k, periodic_z);
+    let pos = |k: usize, j: usize, i: usize| match c {
+        0 => (i as f64 * dx, (j as f64 + 0.5) * dy, (k as f64 + 0.5) * dz),
+        1 => ((i as f64 + 0.5) * dx, j as f64 * dy, (k as f64 + 0.5) * dz),
+        _ => ((i as f64 + 0.5) * dx, (j as f64 + 0.5) * dy, k as f64 * dz),
+    };
+    let index = |k: usize, j: usize, i: usize| match c {
+        0 => g.uface(k, j, i),
+        1 => g.vface(k, j, i),
+        _ => g.wface(k, j, i),
+    };
+    let dirs = |m: usize, f: f64| {
+        if m <= 1 {
+            vec![(0usize, 1.0)]
+        } else {
+            vec![(0, 1.0 - f), (1, f)]
+        }
+    };
+    let mut out = Vec::with_capacity(8);
+    let kz = |dk: usize| if dk == 0 { k0 } else { k1 };
+    for (dk, wk) in dirs(planes_k, fz) {
+        for (dj, wj) in dirs(max_j, fy) {
+            for (di, wi) in dirs(max_i, fx) {
+                let (k, j, i) = (kz(dk), j0 + dj, i0 + di);
+                let (x, y, z) = pos(k, j, i);
+                let idx = index(k, j, i);
+                out.push(StencilNode {
+                    idx,
+                    x,
+                    y,
+                    z,
+                    weight: wi * wj * wk,
+                    zw: wk,
+                    fallback: fallback(idx),
+                });
+            }
+        }
+    }
+    out
+}
+
+/// Weighted least-squares fit of `a + b (x−x0) + c (y−y0) + d (z−z0)`
+/// through `pts` (each `(x, y, z, value, weight)`; zero-weight points drop
+/// out) evaluated at `at`; directions in which every weighted point has
+/// the same coordinate are dropped. `None` if the normal matrix is
+/// singular relative to its scale.
+pub(crate) fn linear_fit(pts_w: &[(f64, f64, f64, f64, f64)], at: (f64, f64, f64)) -> Option<f64> {
+    let pts: Vec<(f64, f64, f64, f64, f64)> = pts_w.iter().copied().filter(|p| p.4 > 0.0).collect();
+    let spread = |f: &dyn Fn(&(f64, f64, f64, f64, f64)) -> f64| {
+        let (lo, hi) = pts
+            .iter()
+            .fold((f64::INFINITY, f64::NEG_INFINITY), |(lo, hi), p| {
+                (lo.min(f(p)), hi.max(f(p)))
+            });
+        hi - lo > 1e-12 * (hi.abs() + lo.abs() + 1e-300)
+    };
+    let on = [true, spread(&|p| p.0), spread(&|p| p.1), spread(&|p| p.2)];
+    let cols: Vec<usize> = (0..4).filter(|&c| on[c]).collect();
+    let m = cols.len();
+    if pts.len() < m {
+        return None;
+    }
+    let mut a = vec![vec![0.0; m + 1]; m];
+    for p in &pts {
+        let full = [1.0, p.0 - at.0, p.1 - at.1, p.2 - at.2];
+        let r: Vec<f64> = cols.iter().map(|&c| full[c]).collect();
+        for i in 0..m {
+            for j in 0..m {
+                a[i][j] += p.4 * r[i] * r[j];
+            }
+            a[i][m] += p.4 * r[i] * p.3;
+        }
+    }
+    let scale: f64 = (0..m).map(|i| a[i][i]).product();
+    if scale <= 0.0 {
+        return None;
+    }
+    let mut det = 1.0;
+    for col in 0..m {
+        let piv = (col..m)
+            .max_by(|&p, &q| a[p][col].abs().partial_cmp(&a[q][col].abs()).unwrap())
+            .unwrap();
+        a.swap(col, piv);
+        let d = a[col][col];
+        det *= d;
+        if d.abs() <= 1e-14 * scale.powf(1.0 / m as f64) {
+            return None;
+        }
+        for r in col + 1..m {
+            let f = a[r][col] / d;
+            for c in col..=m {
+                a[r][c] -= f * a[col][c];
+            }
+        }
+    }
+    if det.abs() <= 1e-10 * scale {
+        return None;
+    }
+    let mut x = vec![0.0; m];
+    for i in (0..m).rev() {
+        let mut s = a[i][m];
+        for j in i + 1..m {
+            s -= a[i][j] * x[j];
+        }
+        x[i] = s / a[i][i];
+    }
+    Some(x[0])
+}
+
+impl Ghost {
+    fn reconstruct(&self, values: &[f64]) -> f64 {
+        let mut pts: Vec<(f64, f64, f64, f64, f64)> = self
+            .nodes
+            .iter()
+            .filter(|n| n.fallback.is_none())
+            .map(|n| (n.x, n.y, n.z, values[n.idx], n.zw))
+            .collect();
+        pts.push((self.foot.0, self.foot.1, self.foot.2, self.u_surface, 1.0));
+        if let Some(val) = linear_fit(&pts, (self.x, self.y, self.z)) {
+            return val;
+        }
+        let mut probe = 0.0;
+        for n in &self.nodes {
+            probe += n.weight * n.fallback.unwrap_or_else(|| values[n.idx]);
+        }
+        self.u_surface + (probe - self.u_surface) * (self.s_face / self.s_probe)
+    }
+}
+
+impl Mask {
+    /// Classify the grid against `body` at `t`. A solid cell on a domain
+    /// side is refused unless that side is a Velocity, SlipWall or Periodic
+    /// side (an outlet may not be blocked).
+    pub fn build(body: &Body, g: Grid, t: f64, b: Boundaries) -> Result<Self, String> {
+        let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
+        let periodic = b.z0 == Side::Periodic;
+        let xc = |i: usize| (i as f64 + 0.5) * dx;
+        let yc = |j: usize| (j as f64 + 0.5) * dy;
+        let zc = |k: usize| (k as f64 + 0.5) * dz;
+        let mut cell_fluid = vec![true; g.cells()];
+        let mut fluid_cells = 0;
+        let mut anchor = None;
+        let allowed = |side: Side| matches!(side, Side::Velocity | Side::Periodic | Side::SlipWall);
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 0..nx {
+                    let fluid = body.phi(xc(i), yc(j), zc(k), t) > 0.0;
+                    let idx = g.cell(k, j, i);
+                    cell_fluid[idx] = fluid;
+                    if fluid {
+                        fluid_cells += 1;
+                        if anchor.is_none() {
+                            anchor = Some(idx);
+                        }
+                    } else {
+                        let touches = (i == 0 && !allowed(b.x0))
+                            || (i + 1 == nx && !allowed(b.x1))
+                            || (j == 0 && !allowed(b.y0))
+                            || (j + 1 == ny && !allowed(b.y1))
+                            || (k == 0 && !allowed(b.z0))
+                            || (k + 1 == nz && !allowed(b.z1));
+                        if touches {
+                            return Err(format!(
+                                "embedded body reaches a domain side that is not a Velocity/Periodic side at cell ({k}, {j}, {i})"
+                            ));
+                        }
+                    }
+                }
+            }
+        }
+        let Some(anchor) = anchor else {
+            return Err("embedded body covers the whole domain".into());
+        };
+        let h_min = dx.min(dy).min(dz);
+        let reach = 1.5 * h_min;
+        let eps = 1e-6 * h_min;
+        let is_fluid = |k: usize, j: usize, i: usize| cell_fluid[g.cell(k, j, i)];
+        let mut u_kind = vec![FaceKind::Fluid; g.n_ufaces()];
+        let mut v_kind = vec![FaceKind::Fluid; g.n_vfaces()];
+        let mut w_kind = vec![FaceKind::Fluid; g.n_wfaces()];
+        let kind_of = |phi: f64| {
+            if phi > -reach {
+                FaceKind::Ghost
+            } else {
+                FaceKind::Solid
+            }
+        };
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 1..nx {
+                    if !(is_fluid(k, j, i - 1) && is_fluid(k, j, i)) {
+                        u_kind[g.uface(k, j, i)] =
+                            kind_of(body.phi(i as f64 * dx, yc(j), zc(k), t));
+                    }
+                }
+            }
+            for j in 1..ny {
+                for i in 0..nx {
+                    if !(is_fluid(k, j - 1, i) && is_fluid(k, j, i)) {
+                        v_kind[g.vface(k, j, i)] =
+                            kind_of(body.phi(xc(i), j as f64 * dy, zc(k), t));
+                    }
+                }
+            }
+        }
+        let w_range = if periodic { 0..nz } else { 1..nz };
+        for k in w_range.clone() {
+            let below = if k > 0 { k - 1 } else { nz - 1 };
+            for j in 0..ny {
+                for i in 0..nx {
+                    if !(is_fluid(below, j, i) && is_fluid(k, j, i)) {
+                        w_kind[g.wface(k, j, i)] =
+                            kind_of(body.phi(xc(i), yc(j), k as f64 * dz, t));
+                    }
+                }
+            }
+        }
+        if periodic {
+            for j in 0..ny {
+                for i in 0..nx {
+                    w_kind[g.wface(nz, j, i)] = w_kind[g.wface(0, j, i)];
+                }
+            }
+        }
+        let face_pos = |c: usize, k: usize, j: usize, i: usize| match c {
+            0 => (i as f64 * dx, yc(j), zc(k)),
+            1 => (xc(i), j as f64 * dy, zc(k)),
+            _ => (xc(i), yc(j), k as f64 * dz),
+        };
+        let kji_of = |c: usize, idx: usize| -> (usize, usize, usize) {
+            match c {
+                0 => (idx / ((nx + 1) * ny), (idx / (nx + 1)) % ny, idx % (nx + 1)),
+                1 => (idx / (nx * (ny + 1)), (idx / nx) % (ny + 1), idx % nx),
+                _ => (idx / (nx * ny), (idx / nx) % ny, idx % nx),
+            }
+        };
+        let build_ghost = |c: usize,
+                           kinds: &[FaceKind],
+                           idx: usize,
+                           (x, y, z): (f64, f64, f64),
+                           flux_sign: f64|
+         -> Ghost {
+            let s_face = body.phi(x, y, z, t);
+            let (n1, n2, n3) = body.normal(x, y, z, t, eps);
+            let foot = (x - s_face * n1, y - s_face * n2, z - s_face * n3);
+            let s_probe = s_face.abs() + h_min;
+            let probe = (
+                foot.0 + s_probe * n1,
+                foot.1 + s_probe * n2,
+                foot.2 + s_probe * n3,
+            );
+            let vel = body.surface_velocity(foot.0, foot.1, foot.2, t);
+            let u_surface = [vel.0, vel.1, vel.2][c];
+            let nodes = stencil_nodes(probe, c, g, periodic, |nidx| {
+                if kinds[nidx] == FaceKind::Fluid {
+                    None
+                } else {
+                    let (kk, jj, ii) = kji_of(c, nidx);
+                    let (px, py, pz) = face_pos(c, kk, jj, ii);
+                    let s = body.phi(px, py, pz, t);
+                    let (m1, m2, m3) = body.normal(px, py, pz, t, eps);
+                    let f = body.surface_velocity(px - s * m1, py - s * m2, pz - s * m3, t);
+                    Some([f.0, f.1, f.2][c])
+                }
+            });
+            Ghost {
+                idx,
+                x,
+                y,
+                z,
+                foot,
+                u_surface,
+                s_face,
+                s_probe,
+                nodes,
+                flux_sign,
+            }
+        };
+        let mut u_ghosts = Vec::new();
+        let mut v_ghosts = Vec::new();
+        let mut w_ghosts = Vec::new();
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 1..nx {
+                    let idx = g.uface(k, j, i);
+                    if u_kind[idx] == FaceKind::Ghost {
+                        let sign = if is_fluid(k, j, i - 1) {
+                            1.0
+                        } else if is_fluid(k, j, i) {
+                            -1.0
+                        } else {
+                            0.0
+                        };
+                        u_ghosts.push(build_ghost(0, &u_kind, idx, face_pos(0, k, j, i), sign));
+                    }
+                }
+            }
+            for j in 1..ny {
+                for i in 0..nx {
+                    let idx = g.vface(k, j, i);
+                    if v_kind[idx] == FaceKind::Ghost {
+                        let sign = if is_fluid(k, j - 1, i) {
+                            1.0
+                        } else if is_fluid(k, j, i) {
+                            -1.0
+                        } else {
+                            0.0
+                        };
+                        v_ghosts.push(build_ghost(1, &v_kind, idx, face_pos(1, k, j, i), sign));
+                    }
+                }
+            }
+        }
+        for k in w_range {
+            let below = if k > 0 { k - 1 } else { nz - 1 };
+            for j in 0..ny {
+                for i in 0..nx {
+                    let idx = g.wface(k, j, i);
+                    if w_kind[idx] == FaceKind::Ghost {
+                        let sign = if is_fluid(below, j, i) {
+                            1.0
+                        } else if is_fluid(k, j, i) {
+                            -1.0
+                        } else {
+                            0.0
+                        };
+                        w_ghosts.push(build_ghost(2, &w_kind, idx, face_pos(2, k, j, i), sign));
+                    }
+                }
+            }
+        }
+        Ok(Self {
+            grid: g,
+            periodic_z: periodic,
+            cell_fluid,
+            u_kind,
+            v_kind,
+            w_kind,
+            u_ghosts,
+            v_ghosts,
+            w_ghosts,
+            anchor,
+            fluid_cells,
+        })
+    }
+
+    #[inline]
+    #[must_use]
+    pub fn is_fluid_cell(&self, idx: usize) -> bool {
+        self.cell_fluid[idx]
+    }
+    #[inline]
+    #[must_use]
+    pub fn u_kind(&self, idx: usize) -> FaceKind {
+        self.u_kind[idx]
+    }
+    #[inline]
+    #[must_use]
+    pub fn v_kind(&self, idx: usize) -> FaceKind {
+        self.v_kind[idx]
+    }
+    #[inline]
+    #[must_use]
+    pub fn w_kind(&self, idx: usize) -> FaceKind {
+        self.w_kind[idx]
+    }
+    #[must_use]
+    pub fn anchor(&self) -> usize {
+        self.anchor
+    }
+    #[must_use]
+    pub fn fluid_cells(&self) -> usize {
+        self.fluid_cells
+    }
+    #[must_use]
+    pub fn ghost_faces(&self) -> usize {
+        self.u_ghosts.len() + self.v_ghosts.len() + self.w_ghosts.len()
+    }
+    #[must_use]
+    pub fn grid(&self) -> Grid {
+        self.grid
+    }
+    #[must_use]
+    pub fn periodic_z(&self) -> bool {
+        self.periodic_z
+    }
+
+    /// Impose the wall on `(u, v, w)` from the same field.
+    pub fn impose(&self, body: &Body, u: &mut [f64], v: &mut [f64], w: &mut [f64], t: f64) -> f64 {
+        let (us, vs, ws) = (u.to_vec(), v.to_vec(), w.to_vec());
+        self.impose_from(body, &us, &vs, &ws, u, v, w, t)
+    }
+
+    /// Solid faces: the surface velocity; ghost faces: the reconstruction
+    /// from the SOURCE field, minus the shared flux compatibility
+    /// correction over the flux-carrying ghosts. Returns the correction.
+    #[allow(clippy::too_many_arguments)]
+    pub fn impose_from(
+        &self,
+        body: &Body,
+        u_src: &[f64],
+        v_src: &[f64],
+        w_src: &[f64],
+        u: &mut [f64],
+        v: &mut [f64],
+        w: &mut [f64],
+        t: f64,
+    ) -> f64 {
+        let g = self.grid;
+        let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
+        for k in 0..nz {
+            for j in 0..ny {
+                for i in 1..nx {
+                    let idx = g.uface(k, j, i);
+                    if self.u_kind[idx] == FaceKind::Solid {
+                        u[idx] = body
+                            .surface_velocity(
+                                i as f64 * dx,
+                                (j as f64 + 0.5) * dy,
+                                (k as f64 + 0.5) * dz,
+                                t,
+                            )
+                            .0;
+                    }
+                }
+            }
+            for j in 1..ny {
+                for i in 0..nx {
+                    let idx = g.vface(k, j, i);
+                    if self.v_kind[idx] == FaceKind::Solid {
+                        v[idx] = body
+                            .surface_velocity(
+                                (i as f64 + 0.5) * dx,
+                                j as f64 * dy,
+                                (k as f64 + 0.5) * dz,
+                                t,
+                            )
+                            .1;
+                    }
+                }
+            }
+        }
+        for k in 0..=nz {
+            for j in 0..ny {
+                for i in 0..nx {
+                    let idx = g.wface(k, j, i);
+                    if self.w_kind[idx] == FaceKind::Solid {
+                        w[idx] = body
+                            .surface_velocity(
+                                (i as f64 + 0.5) * dx,
+                                (j as f64 + 0.5) * dy,
+                                k as f64 * dz,
+                                t,
+                            )
+                            .2;
+                    }
+                }
+            }
+        }
+        let u_vals: Vec<f64> = self
+            .u_ghosts
+            .iter()
+            .map(|gh| gh.reconstruct(u_src))
+            .collect();
+        let v_vals: Vec<f64> = self
+            .v_ghosts
+            .iter()
+            .map(|gh| gh.reconstruct(v_src))
+            .collect();
+        let w_vals: Vec<f64> = self
+            .w_ghosts
+            .iter()
+            .map(|gh| gh.reconstruct(w_src))
+            .collect();
+        let (au, av, aw) = (dy * dz, dx * dz, dx * dy);
+        let mut net = 0.0;
+        let mut area = 0.0;
+        for (gh, &val) in self.u_ghosts.iter().zip(&u_vals) {
+            if gh.flux_sign != 0.0 {
+                net += gh.flux_sign * val * au;
+                area += au;
+            }
+        }
+        for (gh, &val) in self.v_ghosts.iter().zip(&v_vals) {
+            if gh.flux_sign != 0.0 {
+                net += gh.flux_sign * val * av;
+                area += av;
+            }
+        }
+        for (gh, &val) in self.w_ghosts.iter().zip(&w_vals) {
+            if gh.flux_sign != 0.0 {
+                net += gh.flux_sign * val * aw;
+                area += aw;
+            }
+        }
+        let correction = if area > 0.0 { net / area } else { 0.0 };
+        for (gh, &val) in self.u_ghosts.iter().zip(&u_vals) {
+            u[gh.idx] = val - gh.flux_sign * correction;
+        }
+        for (gh, &val) in self.v_ghosts.iter().zip(&v_vals) {
+            v[gh.idx] = val - gh.flux_sign * correction;
+        }
+        for (gh, &val) in self.w_ghosts.iter().zip(&w_vals) {
+            w[gh.idx] = val - gh.flux_sign * correction;
+        }
+        // The periodic seam: the w face at k = nz is the face at k = 0.
+        for j in 0..ny {
+            for i in 0..nx {
+                let (f0, fn_) = (g.wface(0, j, i), g.wface(nz, j, i));
+                if self.w_kind[f0] != FaceKind::Fluid && self.w_kind[fn_] == self.w_kind[f0] {
+                    w[fn_] = w[f0];
+                }
+            }
+        }
+        correction
+    }
+}