rtx-cfd: overset A-P2 — the patch overlaps the background (OversetPisoSolver), gated S1–S5

Background = the embedded solver with a mask from the overlap classification
(embedded/{mod,projection}.rs: module split, projection's solve/apply halves,
set_overlap, fringe p' Dirichlet by elimination into extra_diag/rhs, anchor
dropped, set_inner_stop_factor, phase API begin_step/solve_correction/
apply_correction/end_step; advance rebuilt on the phases — every suite digit-
identical, FSI2 default line-for-line). Patch = the curvilinear solver with an
acceptor ring (set_side_velocity; set_acceptor_ring/stamp_acceptors/
set_acceptor_correction; acceptor Dirichlet by elimination into
PressureSystem.links so the BiCGSTAB stop stays in flux units — identity rows
measured unconverged at 2431 iterations; same phase API). overset/overlap.rs:
OverlapMap — hole/fringe/active from the patch's own indices (hole = body or
k <= nn-1-overlap_rows, DEFAULT_OVERLAP_ROWS = 4 from the 2.9 h depth budget),
dual-quad inverse-bilinear donors patch→fringe, lattice donors →acceptors,
both invariants asserted, mass-defect measures. overset/mod.rs:
OversetPisoSolver — advance (exchange rebuilt BEFORE the predictors from the
previous corrected field), alternating Schwarz on the acceptor p' vector with
Anderson(3) (plain Schwarz measured 0.82/round: floating patch, Neumann wall)
and the previous step's vector as warm start (1 round/corrector at steady
state), stop relative to the STEP's p' scale (the MG absolute stop is
1e-9/dt² in pressure — the whole second correction), set_patch_mesh,
snapshot/restore carrying the warm-start vector.

Gates: overlap linear-exact 1e-13, quadratic orders 1.96/1.99 (acceptors),
1.40/1.91 (fringe); half-couplings: patch with exact acceptors Stokes 2.07/1.98
+ 2.08/1.98, upwind 0.84/0.84, background with exact fringe 7.86e-3/2.90e-3/
1.09e-3 (1.44/1.41); two-mesh MMS n=32/64: background 8.717e-3/4.207e-3 (1.03x/
0.97x the embedded circle), patch 1.322e-2/6.904e-3 (1.5-1.6x), orders 1.05/
0.94, patch div <= 5e-13, overlap mass defect 3.6e-3 -> 8.2e-4 of the overlap
flux (under the registered 1e-3 from n=64; disclosed at 32); motion: stationary
patch through set_patch_mesh bit-identical, snapshot/restore with a pending mesh
bit-identical, translating phantom circle 1.22x/1.19x the static level over
4.5 cells. Inherited, disclosed: poisson_equivalence's no-body multigrid pin
fails by 3.9e-9 at d46fb0b (M1's commit; verified in a clean worktree).

Co-Authored-By: Claude Fable 5.1 <[email protected]>
Claude-Session: https://claude.ai/code/session_01X2GmJXeQ2njUecEKiJZ1G2

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

@@ -0,0 +1,539 @@
+//! The overset (chimera) hybrid: a curvilinear patch around the body over
+//! the fixed background grid (`docs/overset_metal_campaign.md` §2, A-P2).
+//!
+//! [`OversetPisoSolver`] drives an [`EmbeddedPisoSolver`] (the background,
+//! its mask from the overlap classification: holes, fringe) and a
+//! [`CurvilinearPisoSolver`] (the patch, its outer row an acceptor ring)
+//! through one PISO step together:
+//!
+//! 1. patch predictor (`begin_step`: mesh swap if the patch moves, fluxes,
+//!    `u*`, matrix); if the mesh moved, the overlap is rebuilt and the
+//!    fringe re-stamped from the patch's previous corrected field (values
+//!    identical when the mesh did not move — the moving path with a
+//!    stationary patch is the static path to the bit);
+//! 2. background predictor (`begin_step`);
+//! 3. per corrector, alternating Schwarz on `p'` with Dirichlet transmission
+//!    both ways: fringe `p'` ← patch `p'` (zero in the first round),
+//!    background solve; acceptor `p'` ← background `p'`, patch solve; stop
+//!    when the exchanged values change by less than `schwarz_tolerance ×
+//!    max |p'|`; then ONE application of the correction on each mesh;
+//! 4. the exchange for the next step, from the corrected fields (the
+//!    boundary-history principle): prescribed background faces and fringe
+//!    `p` from the patch, acceptor `u, v, p` from the background.
+//!
+//! The overlap mass defect — continuity is enforced on neither the fringe
+//! nor the acceptor cells — is measured every step on both sides.
+
+pub mod overlap;
+
+pub use overlap::{Acceptor, CellClass, DualDonor, FringeEntry, LatticeDonor, OverlapMap};
+
+use crate::error::{CfdError, CfdResult};
+use crate::mesh::PatchMesh;
+use crate::solvers::incompressible::curvilinear::{
+    CurvilinearPisoSolver, CurvilinearSolverState, PatchField,
+};
+use crate::solvers::incompressible::embedded::{EmbeddedPisoSolver, EmbeddedSolverState};
+use crate::solvers::incompressible::flow_field::FlowField;
+
+/// Parameters of the composite step.
+#[derive(Debug, Clone)]
+pub struct OversetParameters {
+    /// Projections per step (both meshes).
+    pub corrector_steps: usize,
+    /// Schwarz stop: the largest change of an exchanged `p'` value between
+    /// rounds, relative to the largest exchanged `|p'|`.
+    pub schwarz_tolerance: f64,
+    /// Round cap per corrector.
+    pub max_rounds: usize,
+    /// Anderson-acceleration depth on the acceptor `p'` vector (0 = plain
+    /// alternating Schwarz). Plain Schwarz converges at ≈ 0.82 per round
+    /// here — the patch's wall is Neumann for `p'`, so its pressure level
+    /// is pinned only through the fringe and decays weakly across the
+    /// overlap (measured: 4.5e-3 relative after 20 rounds).
+    pub anderson_depth: usize,
+    /// Patch rows below the acceptor row kept non-hole
+    /// (`overlap::DEFAULT_OVERLAP_ROWS`).
+    pub overlap_rows: usize,
+}
+
+impl Default for OversetParameters {
+    fn default() -> Self {
+        Self {
+            corrector_steps: 2,
+            schwarz_tolerance: 1e-3,
+            max_rounds: 20,
+            anderson_depth: 3,
+            overlap_rows: overlap::DEFAULT_OVERLAP_ROWS,
+        }
+    }
+}
+
+/// The two fields.
+#[derive(Debug, Clone)]
+pub struct OversetField {
+    /// Background staggered field.
+    pub background: FlowField,
+    /// Patch collocated field.
+    pub patch: PatchField,
+}
+
+/// What one composite step reports.
+#[derive(Debug, Clone)]
+pub struct OversetResult {
+    /// Schwarz rounds per corrector.
+    pub rounds: Vec<usize>,
+    /// Whether every corrector's Schwarz iteration met its stop.
+    pub schwarz_converged: bool,
+    /// Background mass residual after the last corrector (its own measure).
+    pub background_residual: f64,
+    /// Largest patch cell imbalance after the last corrector.
+    pub patch_max_divergence: f64,
+    /// Patch BiCGSTAB iterations, summed.
+    pub patch_poisson_iterations: usize,
+    /// Whether every patch pressure solve converged.
+    pub patch_converged: bool,
+    /// `Σ_fringe |div|` on the background (volume flux).
+    pub background_mass_defect: f64,
+    /// `Σ_acceptors |div|` on the patch (volume flux).
+    pub patch_mass_defect: f64,
+    /// `Σ |F|` over the acceptor–interior faces: the flux scale through the overlap.
+    pub overlap_flux_scale: f64,
+    /// Background cells that changed class in this step's overlap rebuild.
+    pub reclassified_cells: usize,
+    /// Background cells that jumped hole → active (patch moved > 1 cell).
+    pub fresh_cells: usize,
+}
+
+/// Restorable state of the composite (the coupling re-runs a step).
+#[derive(Clone)]
+pub struct OversetSolverState {
+    background: EmbeddedSolverState,
+    patch: CurvilinearSolverState,
+    overlap: OverlapMap,
+    pending: Option<PatchMesh>,
+    acceptor_warm: Vec<f64>,
+}
+
+/// The composite solver.
+pub struct OversetPisoSolver {
+    background: EmbeddedPisoSolver,
+    patch: CurvilinearPisoSolver,
+    overlap: OverlapMap,
+    params: OversetParameters,
+    grid: (usize, usize, f64, f64),
+    pending: Option<PatchMesh>,
+    /// The first corrector's converged acceptor `p'` of the previous step:
+    /// the temporal warm start (the correction is correlated step to step).
+    acceptor_warm: Vec<f64>,
+}
+
+/// Anderson acceleration of a fixed-point iteration `x ← G(x)` with residual
+/// `r = G(x) − x`, depth `m` (least squares by normal equations, tiny).
+struct Anderson {
+    m: usize,
+    xs: Vec<Vec<f64>>,
+    rs: Vec<Vec<f64>>,
+}
+
+impl Anderson {
+    fn new(m: usize) -> Self {
+        Self {
+            m,
+            xs: Vec::new(),
+            rs: Vec::new(),
+        }
+    }
+
+    /// Next iterate from the current `x` and its residual `r`.
+    fn next(&mut self, x: &[f64], r: &[f64]) -> Vec<f64> {
+        self.xs.push(x.to_vec());
+        self.rs.push(r.to_vec());
+        while self.xs.len() > self.m + 1 {
+            self.xs.remove(0);
+            self.rs.remove(0);
+        }
+        let k = self.xs.len() - 1;
+        if self.m == 0 || k == 0 {
+            return x.iter().zip(r).map(|(a, b)| a + b).collect();
+        }
+        // Differences ΔR_i = r_{i+1} − r_i, ΔX_i = x_{i+1} − x_i, i = 0..k.
+        let n = x.len();
+        let dr: Vec<Vec<f64>> = (0..k)
+            .map(|i| (0..n).map(|j| self.rs[i + 1][j] - self.rs[i][j]).collect())
+            .collect();
+        let dx: Vec<Vec<f64>> = (0..k)
+            .map(|i| (0..n).map(|j| self.xs[i + 1][j] - self.xs[i][j]).collect())
+            .collect();
+        // Normal equations (ΔRᵀΔR) γ = ΔRᵀ r, Tikhonov-regularised.
+        let mut a = vec![vec![0.0; k]; k];
+        let mut b = vec![0.0; k];
+        let mut trace = 0.0;
+        for i in 0..k {
+            for l in 0..k {
+                a[i][l] = dr[i].iter().zip(&dr[l]).map(|(p, q)| p * q).sum();
+            }
+            trace += a[i][i];
+            b[i] = dr[i].iter().zip(r).map(|(p, q)| p * q).sum();
+        }
+        for i in 0..k {
+            a[i][i] += 1e-10 * trace.max(1e-300);
+        }
+        let gamma = solve_small(a, b);
+        (0..n)
+            .map(|j| {
+                let mut v = x[j] + r[j];
+                for i in 0..k {
+                    v -= gamma[i] * (dx[i][j] + dr[i][j]);
+                }
+                v
+            })
+            .collect()
+    }
+}
+
+/// Gaussian elimination with partial pivoting on a tiny dense system.
+fn solve_small(mut a: Vec<Vec<f64>>, mut b: Vec<f64>) -> Vec<f64> {
+    let k = b.len();
+    for col in 0..k {
+        let piv = (col..k)
+            .max_by(|&i, &j| a[i][col].abs().partial_cmp(&a[j][col].abs()).unwrap())
+            .unwrap();
+        a.swap(col, piv);
+        b.swap(col, piv);
+        let d = a[col][col];
+        if d.abs() <= 1e-300 {
+            continue;
+        }
+        for row in col + 1..k {
+            let f = a[row][col] / d;
+            for c in col..k {
+                a[row][c] -= f * a[col][c];
+            }
+            b[row] -= f * b[col];
+        }
+    }
+    let mut x = vec![0.0; k];
+    for row in (0..k).rev() {
+        let mut v = b[row];
+        for c in row + 1..k {
+            v -= a[row][c] * x[c];
+        }
+        x[row] = if a[row][row].abs() > 1e-300 {
+            v / a[row][row]
+        } else {
+            0.0
+        };
+    }
+    x
+}
+
+impl OversetPisoSolver {
+    /// Compose a configured background (sides, sources, boundary data) and
+    /// a configured patch (its `Inner` side velocity via
+    /// `set_side_velocity`) on a background of `nx × ny` cells, spacing
+    /// `dx, dy`. Builds the overlap, installs the hole/fringe mask on the
+    /// background and the acceptor ring on the patch.
+    pub fn new(
+        mut background: EmbeddedPisoSolver,
+        mut patch: CurvilinearPisoSolver,
+        grid: (usize, usize, f64, f64),
+        params: OversetParameters,
+    ) -> CfdResult<Self> {
+        let (nx, ny, dx, dy) = grid;
+        let overlap = OverlapMap::build(patch.mesh(), nx, ny, dx, dy, params.overlap_rows)?;
+        background.set_overlap(overlap.background_mask(), overlap.fringe_flags());
+        // Each background solve must be accurate below the Schwarz stop, or
+        // the exchanged values never settle (measured at the default 1e-2).
+        background.set_inner_stop_factor(0.1 * params.schwarz_tolerance);
+        patch.set_acceptor_ring(true);
+        Ok(Self {
+            background,
+            patch,
+            overlap,
+            params,
+            grid,
+            pending: None,
+            acceptor_warm: Vec::new(),
+        })
+    }
+
+    /// The background solver.
+    pub fn background(&self) -> &EmbeddedPisoSolver {
+        &self.background
+    }
+    /// The patch solver.
+    pub fn patch(&self) -> &CurvilinearPisoSolver {
+        &self.patch
+    }
+    /// The current overlap map.
+    pub fn overlap(&self) -> &OverlapMap {
+        &self.overlap
+    }
+    /// Parameters.
+    pub fn parameters(&self) -> &OversetParameters {
+        &self.params
+    }
+    /// The patch's time (the background's agrees).
+    pub fn time(&self) -> f64 {
+        self.patch.time()
+    }
+
+    /// Name the patch geometry the next step ends on (see
+    /// `CurvilinearPisoSolver::set_mesh`; same topology; a fraction of a
+    /// background cell per step).
+    pub fn set_patch_mesh(&mut self, next: PatchMesh) -> CfdResult<()> {
+        self.patch.set_mesh(next.clone())?;
+        self.pending = Some(next);
+        Ok(())
+    }
+
+    /// Stamp both exchanges from the current fields and initialise the
+    /// background (boundary data at the current time). Call once after the
+    /// fields are set.
+    pub fn initialize(&mut self, field: &mut OversetField) -> CfdResult<()> {
+        self.exchange(field);
+        self.background.initialize(&mut field.background)
+    }
+
+    /// The exchange for the next step: prescribed background faces and
+    /// fringe `p` from the patch's cell field; acceptor `u, v, p` from the
+    /// background.
+    fn exchange(&self, field: &mut OversetField) {
+        self.overlap
+            .stamp_fringe_faces(&mut field.background, &field.patch.u, &field.patch.v);
+        let p = self.overlap.fringe_cell_values(&field.patch.p);
+        self.overlap.stamp_fringe_cells(&mut field.background.p, &p);
+        let acc = self
+            .overlap
+            .acceptor_values(&field.background, &field.background.p);
+        self.patch.stamp_acceptors(&mut field.patch, &acc);
+    }
+
+    /// Capture the state.
+    pub fn snapshot(&self) -> OversetSolverState {
+        OversetSolverState {
+            background: self.background.snapshot(),
+            patch: self.patch.snapshot(),
+            overlap: self.overlap.clone(),
+            pending: self.pending.clone(),
+            acceptor_warm: self.acceptor_warm.clone(),
+        }
+    }
+
+    /// Restore a captured state.
+    pub fn restore(&mut self, state: &OversetSolverState) {
+        self.background.restore(&state.background);
+        self.patch.restore(&state.patch);
+        self.overlap = state.overlap.clone();
+        self.pending = state.pending.clone();
+        self.acceptor_warm = state.acceptor_warm.clone();
+        // The background's mask and fringe flags come back with its own
+        // state; its fringe Dirichlet data are transient (set every round).
+    }
+
+    /// Advance both meshes one step of `dt`.
+    pub async fn advance(&mut self, field: &mut OversetField, dt: f64) -> CfdResult<OversetResult> {
+        let (nx, ny, dx, dy) = self.grid;
+
+        // 1. If the patch moves, the overlap follows the NEXT mesh before any
+        //    predictor runs: the background is reclassified (fresh cells
+        //    filled from neighbours), and the fringe is re-stamped from the
+        //    patch's previous CORRECTED cell field through the new donors —
+        //    the same numbers as the static path's exchange when the mesh did
+        //    not move, so a stationary patch through this path is the static
+        //    path to the bit.
+        let mut reclassified = 0usize;
+        let mut fresh = 0usize;
+        if self.pending.take().is_some() {
+            let next = self
+                .patch
+                .next_mesh()
+                .expect("set_mesh named the next mesh");
+            let new = OverlapMap::build(next, nx, ny, dx, dy, self.params.overlap_rows)?;
+            for j in 0..ny {
+                for i in 0..nx {
+                    let (was, now) = (self.overlap.class(j, i), new.class(j, i));
+                    if was != now {
+                        reclassified += 1;
+                        if was == CellClass::Hole && now == CellClass::Active {
+                            fresh += 1;
+                            // Fill from the neighbours that already carry a
+                            // pressure (the embedded solver's fresh-cell rule).
+                            let (mut sum, mut count) = (0.0, 0usize);
+                            for (jj, ii) in [
+                                (j, i + 1),
+                                (j, i.wrapping_sub(1)),
+                                (j + 1, i),
+                                (j.wrapping_sub(1), i),
+                            ] {
+                                if jj < ny
+                                    && ii < nx
+                                    && self.overlap.class(jj, ii) != CellClass::Hole
+                                {
+                                    sum += field.background.p[(jj, ii)];
+                                    count += 1;
+                                }
+                            }
+                            if count > 0 {
+                                field.background.p[(j, i)] = sum / count as f64;
+                            }
+                        }
+                    }
+                }
+            }
+            self.overlap = new;
+            self.background
+                .set_overlap(self.overlap.background_mask(), self.overlap.fringe_flags());
+            self.overlap
+                .stamp_fringe_faces(&mut field.background, &field.patch.u, &field.patch.v);
+            let p = self.overlap.fringe_cell_values(&field.patch.p);
+            self.overlap.stamp_fringe_cells(&mut field.background.p, &p);
+        }
+
+        // 1b. Patch predictor (swaps in the pending mesh).
+        let patch_start = self.patch.begin_step(&mut field.patch, dt)?;
+
+        // 2. Background predictor.
+        let bg_start = self.background.begin_step(&mut field.background, dt)?;
+
+        // 3. Correctors: alternating Schwarz on the acceptor p' vector `a`
+        //    (patch solve with Dirichlet a → fringe p' → background solve →
+        //    G(a)), Anderson-accelerated, warm-started in the first
+        //    corrector from the previous step's converged `a`.
+        let fringe_cells: Vec<(usize, usize)> = self
+            .overlap
+            .fringe_cells
+            .iter()
+            .map(|e| (e.j, e.i))
+            .collect();
+        let n_acc = self.overlap.acceptors.len();
+        let mut rounds = Vec::with_capacity(self.params.corrector_steps);
+        let mut schwarz_converged = true;
+        let mut residual_history = Vec::new();
+        let mut final_residual = f64::INFINITY;
+        let mut patch_iterations = 0usize;
+        let mut patch_converged = true;
+        let trace_rounds = std::env::var("RTX_OVERSET_ROUNDS").is_ok();
+        // The stop is relative to the STEP's pressure-correction scale (the
+        // first corrector's): a later corrector's own p' is a mop-up of the
+        // size of the inner solver's absolute-stop noise (≈ 1e-9 in flux
+        // units is ≈ 1e-9/dt² in pressure — 0.02 here, the whole second
+        // correction), so a stop relative to its own magnitude cannot be
+        // met and would only feed noise to the acceleration.
+        let mut step_scale = 0.0_f64;
+        for corrector in 0..self.params.corrector_steps.max(1) {
+            let mut a: Vec<f64> = if corrector == 0 && self.acceptor_warm.len() == n_acc {
+                self.acceptor_warm.clone()
+            } else {
+                vec![0.0; n_acc]
+            };
+            let mut anderson = Anderson::new(self.params.anderson_depth);
+            let mut patch_pc = vec![0.0; self.patch.mesh().cell_count()];
+            let mut done = false;
+            let mut used = 0usize;
+            for round in 0..self.params.max_rounds.max(1) {
+                used = round + 1;
+                // Patch with Dirichlet a.
+                self.patch.set_acceptor_correction(&a);
+                match self.patch.solve_correction(&field.patch, dt) {
+                    Some((pc, out)) => {
+                        patch_iterations += out.iterations;
+                        patch_converged &= out.converged;
+                        patch_pc = pc;
+                    }
+                    None => patch_pc.iter_mut().for_each(|v| *v = 0.0),
+                }
+                // Background with the fringe p' from the patch.
+                let fringe_vals = self.overlap.fringe_cell_values(&patch_pc);
+                self.background
+                    .set_fringe_correction(&fringe_cells, &fringe_vals);
+                self.background.solve_correction(
+                    &mut field.background,
+                    dt,
+                    corrector == 0 || round > 0,
+                )?;
+                let g = self.overlap.acceptor_scalar(&field.background.p_prime);
+                let r: Vec<f64> = g.iter().zip(&a).map(|(x, y)| x - y).collect();
+                let g_max = g.iter().fold(0.0_f64, |m, v| m.max(v.abs()));
+                step_scale = step_scale.max(g_max);
+                let scale = step_scale.max(1e-300);
+                let change = r.iter().fold(0.0_f64, |m, v| m.max(v.abs()));
+                if trace_rounds {
+                    println!(
+                        "      corrector {corrector} round {round}: |G(a) − a| {change:.3e} / step scale {scale:.3e} = {:.3e} (max|G(a)| {g_max:.3e})",
+                        change / scale
+                    );
+                }
+                if change <= self.params.schwarz_tolerance * scale {
+                    done = true;
+                    break;
+                }
+                let next = anderson.next(&a, &r);
+                // Guard: an extrapolation far beyond the data is noise-driven;
+                // take the plain Schwarz step instead.
+                let next_max = next.iter().fold(0.0_f64, |m, v| m.max(v.abs()));
+                a = if next_max > 4.0 * g_max.max(scale) {
+                    g.clone()
+                } else {
+                    next
+                };
+            }
+            schwarz_converged &= done;
+            rounds.push(used);
+            if corrector == 0 {
+                self.acceptor_warm = a.clone();
+            }
+            // One application on each mesh (the patch with the a it solved).
+            let residual = self.background.apply_correction(&mut field.background, dt);
+            residual_history.push(residual);
+            final_residual = residual;
+            self.patch
+                .apply_correction_pub(&mut field.patch, &patch_pc, dt);
+            if corrector + 1 < self.params.corrector_steps.max(1) {
+                field.background.copy_to_starred();
+            }
+        }
+        let patch_max_div = self.patch.max_divergence_pub(&field.patch.flux);
+
+        // 4. The exchange for the next step, then the clocks.
+        self.exchange(field);
+        let outer = self.overlap.acceptor_outer_velocity(&field.background);
+        let (patch_defect, scale) = self.patch.acceptor_mass_defect(&field.patch, &outer);
+        let bg_defect = self.overlap.background_mass_defect(&field.background);
+        let bg = self.background.end_step(
+            &mut field.background,
+            &bg_start,
+            residual_history,
+            rounds.len(),
+            final_residual,
+        );
+        self.patch.end_step(
+            &patch_start,
+            rounds.len(),
+            patch_max_div,
+            patch_iterations,
+            patch_converged,
+        );
+        if (self.patch.time() - self.background.time()).abs()
+            > 1e-12 * self.patch.time().abs().max(1.0)
+        {
+            return Err(CfdError::invalid_parameter(
+                "overset: background and patch clocks disagree".to_string(),
+            ));
+        }
+        Ok(OversetResult {
+            rounds,
+            schwarz_converged,
+            background_residual: bg.solver_result.final_residual,
+            patch_max_divergence: patch_max_div,
+            patch_poisson_iterations: patch_iterations,
+            patch_converged,
+            background_mass_defect: bg_defect,
+            patch_mass_defect: patch_defect,
+            overlap_flux_scale: scale,
+            reclassified_cells: reclassified,
+            fresh_cells: fresh,
+        })
+    }
+}