Omar SobhandClaude Fable 5.1
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Co-Authored-By: Claude Fable 5.1 <[email protected]>
262 lines
10 KiB
Rust
262 lines
10 KiB
Rust
//! The cut geometry of an embedded body on the grid: φ at the cell
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//! corners; inside every cell the interface is the LINEAR interpolant on a
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//! fixed Kuhn split into six tetrahedra (each face into two triangles along
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//! the same diagonal from both sides), so face apertures and cell volumes
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//! are exact for the interpolant, continuous in the corner values, and
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//! consistent across shared faces — the wall polygon's vector area by
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//! closure (`A_w n_w = −Σ_f A_f n_f`) telescopes exactly over a closed body.
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use super::Grid;
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use super::body::Body;
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/// The cut data of one instant.
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#[derive(Debug, Clone)]
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pub struct CutGeometry {
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pub grid: Grid,
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/// φ at the corners, `(nx + 1) × (ny + 1) × (nz + 1)`.
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pub phi: Vec<f64>,
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/// Fluid area fraction of every u / v / w face (the staggered layouts).
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pub a_u: Vec<f64>,
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pub a_v: Vec<f64>,
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pub a_w: Vec<f64>,
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/// Fluid volume fraction of every cell.
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pub vol: Vec<f64>,
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/// The wall polygon's vector area per cell (outward from the fluid),
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/// `−Σ_f A_f n_f · face area`, in area units.
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pub wall: Vec<[f64; 3]>,
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/// φ at the face centres (the wall distance of a near-wall face).
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pub d_u: Vec<f64>,
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pub d_v: Vec<f64>,
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pub d_w: Vec<f64>,
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}
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impl CutGeometry {
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#[inline]
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fn node(g: Grid, k: usize, j: usize, i: usize) -> usize {
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(k * (g.ny + 1) + j) * (g.nx + 1) + i
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}
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/// Build the cut data of `body` at time `t`.
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pub fn build(body: &Body, grid: Grid, t: f64) -> Self {
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let g = grid;
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let (nx, ny, nz, dx, dy, dz) = (g.nx, g.ny, g.nz, g.dx, g.dy, g.dz);
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let mut phi = vec![0.0; (nx + 1) * (ny + 1) * (nz + 1)];
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for k in 0..=nz {
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for j in 0..=ny {
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for i in 0..=nx {
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phi[Self::node(g, k, j, i)] =
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body.phi(i as f64 * dx, j as f64 * dy, k as f64 * dz, t);
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}
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}
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}
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let corner = |k: usize, j: usize, i: usize| phi[Self::node(g, k, j, i)];
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// Face apertures: a face's two triangles along the diagonal from its
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// (0, 0) to its (1, 1) corner in the face's own (a, b) order — the
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// Kuhn split's diagonals: for an x-face (y, z), a y-face (x, z), a
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// z-face (x, y); the same triangles seen from either cell.
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let tri_area_fraction = |p0: f64, p1: f64, p2: f64| -> f64 {
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let v = [p0, p1, p2];
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let pos = v.iter().filter(|&&q| q >= 0.0).count();
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match pos {
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0 => 0.0,
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3 => 1.0,
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1 => {
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let a = v.iter().position(|&q| q >= 0.0).unwrap();
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let (b, c) = ((a + 1) % 3, (a + 2) % 3);
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(v[a] / (v[a] - v[b])) * (v[a] / (v[a] - v[c]))
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}
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_ => {
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let a = v.iter().position(|&q| q < 0.0).unwrap();
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let (b, c) = ((a + 1) % 3, (a + 2) % 3);
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1.0 - (v[a] / (v[a] - v[b])) * (v[a] / (v[a] - v[c]))
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}
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}
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};
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// Quad corners in (a, b) order: q00, q10, q01, q11; triangles
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// (q00, q10, q11) and (q00, q11, q01).
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let quad_fraction = |q00: f64, q10: f64, q01: f64, q11: f64| -> f64 {
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0.5 * (tri_area_fraction(q00, q10, q11) + tri_area_fraction(q00, q11, q01))
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};
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let mut a_u = vec![0.0; (nx + 1) * ny * nz];
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let mut a_v = vec![0.0; nx * (ny + 1) * nz];
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let mut a_w = vec![0.0; nx * ny * (nz + 1)];
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let mut d_u = vec![0.0; (nx + 1) * ny * nz];
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let mut d_v = vec![0.0; nx * (ny + 1) * nz];
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let mut d_w = vec![0.0; nx * ny * (nz + 1)];
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for k in 0..nz {
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for j in 0..ny {
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for i in 0..=nx {
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// x-face at i: corners (j, k), (j+1, k), (j, k+1), (j+1, k+1)
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let (q00, q10, q01, q11) = (
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corner(k, j, i),
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corner(k, j + 1, i),
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corner(k + 1, j, i),
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corner(k + 1, j + 1, i),
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);
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a_u[g.uface(k, j, i)] = quad_fraction(q00, q10, q01, q11);
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d_u[g.uface(k, j, i)] = 0.25 * (q00 + q10 + q01 + q11);
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}
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}
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}
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for k in 0..nz {
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for j in 0..=ny {
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for i in 0..nx {
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// y-face at j: corners (i, k), (i+1, k), (i, k+1), (i+1, k+1)
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let (q00, q10, q01, q11) = (
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corner(k, j, i),
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corner(k, j, i + 1),
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corner(k + 1, j, i),
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corner(k + 1, j, i + 1),
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);
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a_v[g.vface(k, j, i)] = quad_fraction(q00, q10, q01, q11);
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d_v[g.vface(k, j, i)] = 0.25 * (q00 + q10 + q01 + q11);
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}
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}
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}
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for k in 0..=nz {
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for j in 0..ny {
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for i in 0..nx {
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// z-face at k: corners (i, j), (i+1, j), (i, j+1), (i+1, j+1)
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let (q00, q10, q01, q11) = (
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corner(k, j, i),
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corner(k, j, i + 1),
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corner(k, j + 1, i),
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corner(k, j + 1, i + 1),
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);
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a_w[g.wface(k, j, i)] = quad_fraction(q00, q10, q01, q11);
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d_w[g.wface(k, j, i)] = 0.25 * (q00 + q10 + q01 + q11);
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}
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}
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}
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// Cell volumes by the Kuhn split: the six tetrahedra around the
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// diagonal (0,0,0)–(1,1,1) in unit-cube coordinates.
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let mut vol = vec![0.0; g.cells()];
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let mut wall = vec![[0.0; 3]; g.cells()];
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const KUHN: [[[usize; 3]; 4]; 6] = [
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[[0, 0, 0], [1, 0, 0], [1, 1, 0], [1, 1, 1]],
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[[0, 0, 0], [1, 0, 0], [1, 0, 1], [1, 1, 1]],
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[[0, 0, 0], [0, 1, 0], [1, 1, 0], [1, 1, 1]],
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[[0, 0, 0], [0, 1, 0], [0, 1, 1], [1, 1, 1]],
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[[0, 0, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1]],
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[[0, 0, 0], [0, 0, 1], [0, 1, 1], [1, 1, 1]],
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];
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for k in 0..nz {
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for j in 0..ny {
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for i in 0..nx {
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let mut fluid = 0.0;
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for tet in &KUHN {
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let pts: Vec<[f64; 3]> = tet
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.iter()
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.map(|c| [c[0] as f64, c[1] as f64, c[2] as f64])
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.collect();
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let vals: Vec<f64> = tet
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.iter()
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.map(|c| corner(k + c[2], j + c[1], i + c[0]))
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.collect();
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fluid += tet_fluid_volume(&pts, &vals);
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}
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// The six tets fill the unit cube (volume 1).
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let idx = g.cell(k, j, i);
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vol[idx] = fluid;
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let ax = dy * dz;
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let ay = dx * dz;
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let az = dx * dy;
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// Outward normals of the cell's faces times their fluid area,
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// summed; the wall closes the fluid part of the cell.
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let sx = (a_u[g.uface(k, j, i + 1)] - a_u[g.uface(k, j, i)]) * ax;
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let sy = (a_v[g.vface(k, j + 1, i)] - a_v[g.vface(k, j, i)]) * ay;
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let sz = (a_w[g.wface(k + 1, j, i)] - a_w[g.wface(k, j, i)]) * az;
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wall[idx] = [-sx, -sy, -sz];
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}
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}
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}
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Self {
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grid,
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phi,
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a_u,
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a_v,
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a_w,
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vol,
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wall,
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d_u,
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d_v,
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d_w,
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}
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}
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/// Total fluid volume.
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#[must_use]
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pub fn fluid_volume(&self) -> f64 {
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let g = self.grid;
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self.vol.iter().sum::<f64>() * g.dx * g.dy * g.dz
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}
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/// Σ over cells of |wall vector area| (the wall's area up to the
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/// non-planarity of the per-cell polygon) and the closure vector Σ wall.
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#[must_use]
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pub fn wall_area_and_closure(&self) -> (f64, [f64; 3]) {
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let mut area = 0.0;
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let mut sum = [0.0; 3];
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for w in &self.wall {
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area += (w[0] * w[0] + w[1] * w[1] + w[2] * w[2]).sqrt();
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sum[0] += w[0];
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sum[1] += w[1];
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sum[2] += w[2];
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}
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(area, sum)
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}
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}
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fn det3(a: [f64; 3], b: [f64; 3], c: [f64; 3]) -> f64 {
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a[0] * (b[1] * c[2] - b[2] * c[1]) - a[1] * (b[0] * c[2] - b[2] * c[0])
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+ a[2] * (b[0] * c[1] - b[1] * c[0])
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}
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fn tet_volume(p: [[f64; 3]; 4]) -> f64 {
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let e = |a: [f64; 3], b: [f64; 3]| [b[0] - a[0], b[1] - a[1], b[2] - a[2]];
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det3(e(p[0], p[1]), e(p[0], p[2]), e(p[0], p[3])).abs() / 6.0
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}
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fn lerp(a: [f64; 3], b: [f64; 3], t: f64) -> [f64; 3] {
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[
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a[0] + t * (b[0] - a[0]),
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a[1] + t * (b[1] - a[1]),
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a[2] + t * (b[2] - a[2]),
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]
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}
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/// The volume of `{φ ≥ 0}` in a tetrahedron with the linear interpolant of
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/// the corner values `v` (φ = 0 at a corner counts as fluid). Continuous in
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/// `v`: every case's cut points move continuously and the cases agree on
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/// their boundaries.
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fn tet_fluid_volume(pts: &[[f64; 3]], v: &[f64]) -> f64 {
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let total = tet_volume([pts[0], pts[1], pts[2], pts[3]]);
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let pos: Vec<usize> = (0..4).filter(|&q| v[q] >= 0.0).collect();
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let neg: Vec<usize> = (0..4).filter(|&q| v[q] < 0.0).collect();
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let cut = |a: usize, b: usize| lerp(pts[a], pts[b], v[a] / (v[a] - v[b]));
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match pos.len() {
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0 => 0.0,
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4 => total,
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1 => {
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let a = pos[0];
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let (b, c, d) = (neg[0], neg[1], neg[2]);
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tet_volume([pts[a], cut(a, b), cut(a, c), cut(a, d)])
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}
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3 => {
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let a = neg[0];
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let (b, c, d) = (pos[0], pos[1], pos[2]);
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total - tet_volume([pts[a], cut(a, b), cut(a, c), cut(a, d)])
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}
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_ => {
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// Two fluid corners A, B; the fluid wedge {A, B, P_AC, P_AD, P_BC, P_BD}
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// as three tetrahedra.
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let (a, b) = (pos[0], pos[1]);
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let (c, d) = (neg[0], neg[1]);
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let (pac, pad, pbc, pbd) = (cut(a, c), cut(a, d), cut(b, c), cut(b, d));
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tet_volume([pts[a], pts[b], pbc, pbd])
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+ tet_volume([pts[a], pac, pbc, pbd])
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+ tet_volume([pts[a], pac, pad, pbd])
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}
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}
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}
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