R5 design pass: the curved probe's least-squares reconstruction instrument (fit_terms; RTX_E3_FIT_ORDER=3 cubic) — the candidate cut-face closure's flux truncation against the exact box integrals

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

crates/specialized/rtx-cfd/tests/embedded3_wall_position_curved.rs

@@ -8,10 +8,10 @@
 //! exact p(r) (the S2-7b form). `RTX_E3_CURVED_MODE=rigid` is the
 //! linear-exactness mode: solid-body rotation of both cylinders
 //! (`u = Ω r e_θ`, `p = ρ Ω² r²/2`).
-use rtx_cfd::solvers::incompressible::ConvectionScheme;
 use rtx_cfd::solvers::incompressible::embedded3::{
     Body, Boundaries, FaceKind, Field, Fluid, Grid, Parameters, Side, Solver, WallScheme,
 };
+use rtx_cfd::solvers::incompressible::ConvectionScheme;
 
 const MU: f64 = 0.1;
 const RHO: f64 = 1.0;
@@ -447,7 +447,7 @@ fn probe(n: usize, rigid: bool) {
     let is_cut = |c: usize| mask.vol(c) < 1.0 - 1e-9;
     // [wall][band]: bands α<¼, ¼–½, ½–¾, ¾–1, full next to cut, interior;
     // per face [total, viscous group, inertial group] / a_max.
-    let mut acc: [[Vec<[f64; 9]>; 6]; 2] = Default::default();
+    let mut acc: [[Vec<[f64; 11]>; 6]; 2] = Default::default();
     // R5 design pass: the EXACT viscous line integrals over the momentum
     // control volume of a face — the open parts of its four sides (the
     // side-diffusion group: diffusion + solid exchange) and the wall arc
@@ -472,7 +472,8 @@ fn probe(n: usize, rigid: bool) {
         let r = r_of(x, y);
         (R1..=R2).contains(&r)
     };
-    let side = |c: usize, x0: f64, y0: f64, x1: f64, y1: f64, n: [f64; 2]| -> f64 {
+    type GradFn<'a> = &'a dyn Fn(f64, f64) -> [f64; 2];
+    let side = |x0: f64, y0: f64, x1: f64, y1: f64, n: [f64; 2], gf: GradFn| -> f64 {
         let m = 4096;
         let len = ((x1 - x0).powi(2) + (y1 - y0).powi(2)).sqrt();
         let mut s = 0.0;
@@ -480,13 +481,13 @@ fn probe(n: usize, rigid: bool) {
             let t = (q as f64 + 0.5) / m as f64;
             let (x, y) = (x0 + t * (x1 - x0), y0 + t * (y1 - y0));
             if in_gap(x, y) {
-                let g = grad(c, x, y);
+                let g = gf(x, y);
                 s += (g[0] * n[0] + g[1] * n[1]) * len / m as f64;
             }
         }
         MU * s
     };
-    let arc = |c: usize, xa: f64, xb: f64, ya: f64, yb: f64| -> f64 {
+    let arc = |xa: f64, xb: f64, ya: f64, yb: f64, gf: GradFn| -> f64 {
         let mut s = 0.0;
         let m = 1 << 20;
         for (r, sign) in [(R1, -1.0), (R2, 1.0)] {
@@ -494,7 +495,7 @@ fn probe(n: usize, rigid: bool) {
                 let th = std::f64::consts::TAU * (q as f64 + 0.5) / m as f64;
                 let (x, y) = (CENTRE.0 + r * th.cos(), CENTRE.1 + r * th.sin());
                 if x >= xa && x < xb && y >= ya && y < yb {
-                    let g = grad(c, x, y);
+                    let g = gf(x, y);
                     let n = [sign * th.cos(), sign * th.sin()];
                     s += (g[0] * n[0] + g[1] * n[1]) * r * std::f64::consts::TAU / m as f64;
                 }
@@ -502,17 +503,173 @@ fn probe(n: usize, rigid: bool) {
         }
         MU * s
     };
-    // (side-diffusion, wall-shear) exact accelerations for the face of
+    // (side-diffusion, wall-shear) accelerations for the face of
-    // component `c` whose CV is the box [xa, xb] × [ya, yb], aperture `a`.
+    // component `c` whose CV is the box [xa, xb] × [ya, yb], aperture `a`,
-    let exact_terms = |c: usize, xa: f64, xb: f64, ya: f64, yb: f64, a: f64| -> (f64, f64) {
+    // from a gradient field `gf`.
-        let d = side(c, xa, ya, xa, yb, [-1.0, 0.0])
+    let box_terms = |xa: f64, xb: f64, ya: f64, yb: f64, a: f64, gf: GradFn| -> (f64, f64) {
-            + side(c, xb, ya, xb, yb, [1.0, 0.0])
+        let d = side(xa, ya, xa, yb, [-1.0, 0.0], gf)
-            + side(c, xa, ya, xb, ya, [0.0, -1.0])
+            + side(xb, ya, xb, yb, [1.0, 0.0], gf)
-            + side(c, xa, yb, xb, yb, [0.0, 1.0]);
+            + side(xa, ya, xb, ya, [0.0, -1.0], gf)
-        let w = arc(c, xa, xb, ya, yb);
+            + side(xa, yb, xb, yb, [0.0, 1.0], gf);
+        let w = arc(xa, xb, ya, yb, gf);
         let v_eff = a.max(0.1) * h * h * h;
         (d * h / (RHO * v_eff), w * h / (RHO * v_eff))
     };
+    let exact_terms = |c: usize, xa: f64, xb: f64, ya: f64, yb: f64, a: f64| -> (f64, f64) {
+        box_terms(xa, xb, ya, yb, a, &|x, y| grad(c, x, y))
+    };
+    // R5-b candidate at the reconstruction level: a weighted least-squares
+    // QUADRATIC of the exact face values at the open neighbours' centroids
+    // (a 5 × 5 face neighbourhood) with the wall's velocity at three points
+    // of the arc inside the box as strong constraints; its gradient gives
+    // the same box fluxes. Its error against the exact integrals is the
+    // truncation a second-order cut-face closure would leave.
+    let fit_terms =
+        |c: usize, i: usize, j: usize, xa: f64, xb: f64, ya: f64, yb: f64, a: f64| -> (f64, f64) {
+            let mut pts: Vec<(f64, f64, f64, f64)> = Vec::new();
+            for dj in -2i64..=2 {
+                for di in -2i64..=2 {
+                    let (ii, jj) = (i as i64 + di, j as i64 + dj);
+                    if ii < 0 || jj < 0 {
+                        continue;
+                    }
+                    let (ii, jj) = (ii as usize, jj as usize);
+                    let (ok, f, x, y) = if c == 0 {
+                        if ii > nx || jj >= ny {
+                            continue;
+                        }
+                        let f = g.uface(0, jj, ii);
+                        (
+                            mask.u_kind(f) == FaceKind::Fluid && mask.a_u(f) > 0.0,
+                            f,
+                            ii as f64 * h,
+                            (jj as f64 + 0.5) * h,
+                        )
+                    } else {
+                        if ii >= nx || jj > ny {
+                            continue;
+                        }
+                        let f = g.vface(0, jj, ii);
+                        (
+                            mask.v_kind(f) == FaceKind::Fluid && mask.a_v(f) > 0.0,
+                            f,
+                            (ii as f64 + 0.5) * h,
+                            jj as f64 * h,
+                        )
+                    };
+                    if !ok {
+                        continue;
+                    }
+                    let t = &tables[c][3 * f..3 * f + 3];
+                    let val = if c == 0 {
+                        field.u_old[f]
+                    } else {
+                        field.v_old[f]
+                    };
+                    pts.push((x + t[0], y + t[1], val, 1.0));
+                }
+            }
+            // The wall constraints: three points of each arc inside the box.
+            for (r, _) in [(R1, -1.0), (R2, 1.0)] {
+                let m = 1 << 14;
+                let mut ths: Vec<f64> = Vec::new();
+                for q in 0..m {
+                    let th = std::f64::consts::TAU * (q as f64 + 0.5) / m as f64;
+                    let (x, y) = (CENTRE.0 + r * th.cos(), CENTRE.1 + r * th.sin());
+                    if x >= xa && x < xb && y >= ya && y < yb {
+                        ths.push(th);
+                    }
+                }
+                if ths.is_empty() {
+                    continue;
+                }
+                let (t0, t1) = (ths[0], ths[ths.len() - 1]);
+                for th in [t0, 0.5 * (t0 + t1), t1] {
+                    let (x, y) = (CENTRE.0 + r * th.cos(), CENTRE.1 + r * th.sin());
+                    let v = ex.velocity(x, y, rigid);
+                    pts.push((x, y, if c == 0 { v.0 } else { v.1 }, 1.0e4));
+                }
+            }
+            let (xc, yc) = (0.5 * (xa + xb), 0.5 * (ya + yb));
+            // `RTX_E3_FIT_ORDER=3`: a cubic (10 terms) instead of the quadratic (6).
+            let nb = if std::env::var("RTX_E3_FIT_ORDER").is_ok_and(|v| v == "3") {
+                10
+            } else {
+                6
+            };
+            let basis = |x: f64, y: f64| -> [f64; 10] {
+                let (u, v) = ((x - xc) / h, (y - yc) / h);
+                [
+                    1.0,
+                    u,
+                    v,
+                    u * u,
+                    u * v,
+                    v * v,
+                    u * u * u,
+                    u * u * v,
+                    u * v * v,
+                    v * v * v,
+                ]
+            };
+            let mut m = [[0.0f64; 11]; 10];
+            for &(x, y, val, w0) in &pts {
+                let b = basis(x, y);
+                let rho2 = ((x - xc) / h).powi(2) + ((y - yc) / h).powi(2);
+                let w = w0 / (1.0 + rho2);
+                for r in 0..nb {
+                    for cc in 0..nb {
+                        m[r][cc] += w * b[r] * b[cc];
+                    }
+                    m[r][10] += w * b[r] * val;
+                }
+            }
+            // Gaussian elimination with partial pivoting.
+            for k in 0..nb {
+                let piv = (k..nb)
+                    .max_by(|&p1, &p2| m[p1][k].abs().partial_cmp(&m[p2][k].abs()).unwrap())
+                    .unwrap();
+                m.swap(k, piv);
+                let d = m[k][k];
+                if d.abs() < 1e-300 {
+                    continue;
+                }
+                for r in 0..nb {
+                    if r != k {
+                        let fct = m[r][k] / d;
+                        for cc in k..11 {
+                            m[r][cc] -= fct * m[k][cc];
+                        }
+                    }
+                }
+            }
+            let mut coef = [0.0f64; 10];
+            for k in 0..nb {
+                if m[k][k].abs() >= 1e-300 {
+                    coef[k] = m[k][10] / m[k][k];
+                }
+            }
+            let gf = move |x: f64, y: f64| -> [f64; 2] {
+                let (u, v) = ((x - xc) / h, (y - yc) / h);
+                [
+                    (coef[1]
+                        + 2.0 * coef[3] * u
+                        + coef[4] * v
+                        + 3.0 * coef[6] * u * u
+                        + 2.0 * coef[7] * u * v
+                        + coef[8] * v * v)
+                        / h,
+                    (coef[2]
+                        + coef[4] * u
+                        + 2.0 * coef[5] * v
+                        + coef[7] * u * u
+                        + 2.0 * coef[8] * u * v
+                        + 3.0 * coef[9] * v * v)
+                        / h,
+                ]
+            };
+            box_terms(xa, xb, ya, yb, a, &gf)
+        };
     let bin_of = |a: f64, near: bool| -> Option<usize> {
         if a < 1.0 {
             Some(((a * 4.0).floor() as usize).min(3))
@@ -525,7 +682,7 @@ fn probe(n: usize, rigid: bool) {
     // [total, viscous group, inertial group, diffusion, wall shear, solid exchange] / a_max.
     // …, then the side-diffusion error (diff + exch − exact), the wall-shear
     // error (shear − exact) and the exact sum (the quadrature's own error).
-    let groups = |t: &[f64; 7], total: f64, exact: (f64, f64)| -> [f64; 9] {
+    let groups = |t: &[f64; 7], total: f64, exact: (f64, f64), fit: (f64, f64)| -> [f64; 11] {
         [
             total / a_max,
             (t[1] + t[2] + t[5]) / a_max,
@@ -536,6 +693,8 @@ fn probe(n: usize, rigid: bool) {
             (t[1] + t[5] - exact.0) / a_max,
             (t[2] - exact.1) / a_max,
             (exact.0 + exact.1) / a_max,
+            (fit.0 - exact.0) / a_max,
+            (fit.1 - exact.1) / a_max,
         ]
     };
     let wall_of = |x: f64, y: f64| usize::from(r_of(x, y) >= 0.5 * (R1 + R2));
@@ -551,18 +710,23 @@ fn probe(n: usize, rigid: bool) {
             };
             let star = field.u[f] + (dt / RHO) * mask.grad_weight(0, f) * (pp[cp] - pp[cm]) / h;
             let t = terms[0].get(f).copied().unwrap_or([0.0; 7]);
-            let exact = exact_terms(
+            let (xa, xb, ya, yb) = (
-                0,
                 (i as f64 - 0.5) * h,
                 (i as f64 + 0.5) * h,
                 j as f64 * h,
                 (j as f64 + 1.0) * h,
-                mask.a_u(f),
             );
+            let exact = exact_terms(0, xa, xb, ya, yb, mask.a_u(f));
+            let fit = if b < 5 {
+                fit_terms(0, i, j, xa, xb, ya, yb, mask.a_u(f))
+            } else {
+                exact
+            };
             acc[wall_of(i as f64 * h, (j as f64 + 0.5) * h)][b].push(groups(
                 &t,
                 (star - field.u_old[f]) / dt,
                 exact,
+                fit,
             ));
         }
     }
@@ -578,18 +742,23 @@ fn probe(n: usize, rigid: bool) {
             };
             let star = field.v[f] + (dt / RHO) * mask.grad_weight(1, f) * (pp[cp] - pp[cm]) / h;
             let t = terms[1].get(f).copied().unwrap_or([0.0; 7]);
-            let exact = exact_terms(
+            let (xa, xb, ya, yb) = (
-                1,
                 i as f64 * h,
                 (i as f64 + 1.0) * h,
                 (j as f64 - 0.5) * h,
                 (j as f64 + 0.5) * h,
-                mask.a_v(f),
             );
+            let exact = exact_terms(1, xa, xb, ya, yb, mask.a_v(f));
+            let fit = if b < 5 {
+                fit_terms(1, i, j, xa, xb, ya, yb, mask.a_v(f))
+            } else {
+                exact
+            };
             acc[wall_of((i as f64 + 0.5) * h, j as f64 * h)][b].push(groups(
                 &t,
                 (star - field.v_old[f]) / dt,
                 exact,
+                fit,
             ));
         }
     }
@@ -623,7 +792,7 @@ fn probe(n: usize, rigid: bool) {
             pn[w] += 1;
         }
     }
-    let rms = |v: &[[f64; 9]], k: usize| {
+    let rms = |v: &[[f64; 11]], k: usize| {
         (v.iter().map(|x| x[k] * x[k]).sum::<f64>() / v.len().max(1) as f64).sqrt()
     };
     let names = ["α<¼", "¼–½", "½–¾", "¾–1", "full next to cut", "interior"];
@@ -638,7 +807,7 @@ fn probe(n: usize, rigid: bool) {
         let mut line = format!("  probe {mode} n {n} {wname}:");
         for (b, name) in names.iter().enumerate() {
             line += &format!(
-                " {name} {} rms {:.3e} (visc {:.3e} inert {:.3e}; diff {:.3e} shear {:.3e} exch {:.3e}; ERR sides {:.3e} wall {:.3e} quad {:.1e});",
+                " {name} {} rms {:.3e} (visc {:.3e} inert {:.3e}; diff {:.3e} shear {:.3e} exch {:.3e}; ERR sides {:.3e} wall {:.3e} quad {:.1e}; FIT sides {:.3e} wall {:.3e});",
                 acc[w][b].len(),
                 rms(&acc[w][b], 0),
                 rms(&acc[w][b], 1),
@@ -648,7 +817,9 @@ fn probe(n: usize, rigid: bool) {
                 rms(&acc[w][b], 5),
                 rms(&acc[w][b], 6),
                 rms(&acc[w][b], 7),
-                rms(&acc[w][b], 8)
+                rms(&acc[w][b], 8),
+                rms(&acc[w][b], 9),
+                rms(&acc[w][b], 10)
             );
         }
         line += &format!(