Omar SobhandClaude Fable 5.1
2c94ae0bb1
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Co-Authored-By: Claude Fable 5.1 <[email protected]>
443 lines
18 KiB
Rust
443 lines
18 KiB
Rust
//! The two load routes of the 2D solver in 3D. Route A, surface stress
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//! reconstruction: at each surface sample the pressure is linearly
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//! extrapolated to the wall from probes at `h` and `2h` along the normal;
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//! the wall-normal derivatives of the normal and two tangential velocity
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//! components from the same probes with the surface velocity at the wall
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//! (quadratic fit); the tangential derivatives of the normal velocity from
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//! the surface-velocity function; `σ·n = (−p + 2μ ∂ₙuₙ) n + μ Σₜ (∂ₙuₜ +
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//! ∂ₜuₙ) t`. Route B, a momentum balance over a box of whole cells
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//! enclosing the body — it reads no near-wall value; the two are unrelated
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//! readings of one solution.
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use super::body::Body;
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use super::field::Field;
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use super::wall::{FaceKind, Mask, linear_fit, stencil_nodes, z_planes};
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#[derive(Debug, Clone, Copy, PartialEq)]
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pub struct SurfaceForce {
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pub f: [f64; 3],
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pub samples: usize,
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pub skipped: usize,
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}
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fn cross(a: [f64; 3], b: [f64; 3]) -> [f64; 3] {
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[
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a[1] * b[2] - a[2] * b[1],
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a[2] * b[0] - a[0] * b[2],
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a[0] * b[1] - a[1] * b[0],
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]
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}
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fn dot(a: [f64; 3], b: [f64; 3]) -> f64 {
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a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
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}
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/// Two unit tangents orthonormal to `n`.
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fn tangents(n: [f64; 3]) -> ([f64; 3], [f64; 3]) {
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let a = if n[0].abs() < 0.9 {
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[1.0, 0.0, 0.0]
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} else {
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[0.0, 1.0, 0.0]
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};
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let t1 = cross(n, a);
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let l = dot(t1, t1).sqrt();
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let t1 = [t1[0] / l, t1[1] / l, t1[2] / l];
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(t1, cross(n, t1))
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}
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impl Mask {
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/// Pressure at a point from the cell centres: trilinear when the
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/// surrounding cells are fluid, else the least-squares fit through the
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/// fluid ones; `None` if degenerate.
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pub fn pressure_at(&self, p: &[f64], x: f64, y: f64, z: f64) -> Option<f64> {
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let g = self.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 (gx, gy, gz) = (x / dx - 0.5, y / dy - 0.5, z / dz - 0.5);
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let i0 = gx.floor().clamp(0.0, (nx - 2) as f64) as usize;
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let j0 = gy.floor().clamp(0.0, (ny - 2) as f64) as usize;
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let fx = (gx - i0 as f64).clamp(0.0, 1.0);
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let fy = (gy - j0 as f64).clamp(0.0, 1.0);
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let (k0, k1, fz) = z_planes(gz, nz, self.periodic_z());
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// A query on a single weighted plane lives on it (the fit has no
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// z variation to determine).
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let z = if nz == 1 || fz == 0.0 {
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(k0 as f64 + 0.5) * dz
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} else if fz == 1.0 {
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(k1 as f64 + 0.5) * dz
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} else {
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z
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};
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let dirs = |m: usize, f: f64| {
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if m <= 1 {
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vec![(0usize, 1.0)]
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} else {
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vec![(0, 1.0 - f), (1, f)]
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}
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};
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let kz = |dk: usize| if dk == 0 { k0 } else { k1 };
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let mut nodes = Vec::with_capacity(8);
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for (dk, wk) in dirs(nz, fz) {
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for (dj, wj) in dirs(ny, fy) {
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for (di, wi) in dirs(nx, fx) {
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nodes.push((g.cell(kz(dk), j0 + dj, i0 + di), wi * wj * wk));
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}
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}
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}
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// A virtually merged small cell carries its master's pressure
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// unknown, not its own: it is dropped from the fit.
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let usable = |idx: usize| self.is_fluid_cell(idx) && self.master(idx).is_none();
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if nodes.iter().all(|&(idx, _)| usable(idx)) {
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return Some(nodes.iter().map(|&(idx, w)| w * p[idx]).sum());
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}
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// Weighted by the z-direction weight (a z-invariant field then fits
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// as the 2D four-cell fit); the z weight of node `(dk, dj, di)` is
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// `wk`, recovered from the trilinear product.
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let zw = |wk: f64| wk;
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let mut pts: Vec<(f64, f64, f64, f64, f64)> = Vec::new();
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let mut it = nodes.iter();
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for (dk, wk) in dirs(nz, fz) {
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for _ in 0..dirs(ny, fy).len() * dirs(nx, fx).len() {
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let &(idx, _) = it.next().expect("node");
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let _ = dk;
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if usable(idx) {
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let (k, j, i) = g.kji(idx);
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pts.push((
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(i as f64 + 0.5) * dx,
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(j as f64 + 0.5) * dy,
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(k as f64 + 0.5) * dz,
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p[idx],
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zw(wk),
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));
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}
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}
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}
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linear_fit(&pts, (x, y, z))
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}
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/// Velocity at a point: trilinear over a component's nodes when all are
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/// fluid faces, else the fit through the fluid ones and the point's own
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/// boundary intercept with its surface velocity.
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pub fn velocity_at(
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&self,
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body: &Body,
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f: &Field,
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x: f64,
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y: f64,
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z: f64,
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t: f64,
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) -> Option<[f64; 3]> {
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let g = self.grid();
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let eps = 1e-6 * g.dx.min(g.dy).min(g.dz);
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let s = body.phi(x, y, z, t);
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let (n1, n2, n3) = body.normal(x, y, z, t, eps);
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let foot = (x - s * n1, y - s * n2, z - s * n3);
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let vf = body.surface_velocity(foot.0, foot.1, foot.2, t);
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let vel_foot = [vf.0, vf.1, vf.2];
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let mut out = [0.0; 3];
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for c in 0..3 {
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// A query on a single weighted plane of this component's nodes
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// lives on it (the fit has no z variation to determine).
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let (planes, offset) = if c == 2 {
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(if self.periodic_z() { g.nz } else { g.nz + 1 }, 0.0)
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} else {
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(g.nz, 0.5)
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};
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let gz = z / g.dz - offset;
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let (k0, k1, fz) = z_planes(gz, planes, self.periodic_z());
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let zq = if planes <= 1 || fz == 0.0 {
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(k0 as f64 + offset) * g.dz
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} else if fz == 1.0 {
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(k1 as f64 + offset) * g.dz
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} else {
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z
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};
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let foot_c = (foot.0, foot.1, if zq == z { foot.2 } else { zq });
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let nodes = stencil_nodes((x, y, zq), c, g, self.periodic_z(), |_| None);
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let values: &[f64] = [&f.u, &f.v, &f.w][c];
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let fluid = |idx: usize| match c {
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0 => self.u_kind(idx) == FaceKind::Fluid,
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1 => self.v_kind(idx) == FaceKind::Fluid,
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_ => self.w_kind(idx) == FaceKind::Fluid,
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};
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if nodes.iter().all(|n| fluid(n.idx)) {
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out[c] = nodes.iter().map(|n| n.weight * values[n.idx]).sum();
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} else {
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let mut pts: Vec<(f64, f64, f64, f64, f64)> = nodes
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.iter()
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.filter(|n| fluid(n.idx))
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.map(|n| (n.x, n.y, n.z, values[n.idx], n.zw))
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.collect();
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pts.push((foot_c.0, foot_c.1, foot_c.2, vel_foot[c], 1.0));
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out[c] = linear_fit(&pts, (x, y, zq))?;
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}
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}
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Some(out)
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}
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/// The reconstructed traction at one surface point with outward normal `n`.
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pub fn traction_at(
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&self,
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body: &Body,
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f: &Field,
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mu: f64,
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t: f64,
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x: [f64; 3],
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n: [f64; 3],
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) -> Option<[f64; 3]> {
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let g = self.grid();
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let h = g.dx.min(g.dy).min(g.dz);
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let (d1, d2) = (h, 2.0 * h);
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let at = |d: f64| [x[0] + d * n[0], x[1] + d * n[1], x[2] + d * n[2]];
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let (x1, x2) = (at(d1), at(d2));
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let p1 = self.pressure_at(&f.p, x1[0], x1[1], x1[2])?;
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let p2 = self.pressure_at(&f.p, x2[0], x2[1], x2[2])?;
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let u1 = self.velocity_at(body, f, x1[0], x1[1], x1[2], t)?;
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let u2 = self.velocity_at(body, f, x2[0], x2[1], x2[2], t)?;
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let p_wall = p1 + (p1 - p2) * d1 / (d2 - d1);
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let (t1, t2) = tangents(n);
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let us = body.surface_velocity(x[0], x[1], x[2], t);
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let us = [us.0, us.1, us.2];
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let wall_gradient =
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|f1: f64, f2: f64| (f1 * d2 * d2 - f2 * d1 * d1) / (d1 * d2 * (d2 - d1));
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let dn =
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|dir: [f64; 3]| wall_gradient(dot(u1, dir) - dot(us, dir), dot(u2, dir) - dot(us, dir));
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let eps = 1e-6 * h;
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let dt_un = |dir: [f64; 3]| {
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let p = body.surface_velocity(
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x[0] + eps * dir[0],
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x[1] + eps * dir[1],
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x[2] + eps * dir[2],
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t,
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);
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let m = body.surface_velocity(
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x[0] - eps * dir[0],
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x[1] - eps * dir[1],
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x[2] - eps * dir[2],
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t,
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);
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((p.0 - m.0) * n[0] + (p.1 - m.1) * n[1] + (p.2 - m.2) * n[2]) / (2.0 * eps)
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};
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let traction_n = -p_wall + 2.0 * mu * dn(n);
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let traction_t1 = mu * (dn(t1) + dt_un(t1));
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let traction_t2 = mu * (dn(t2) + dt_un(t2));
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Some([
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traction_n * n[0] + traction_t1 * t1[0] + traction_t2 * t2[0],
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traction_n * n[1] + traction_t1 * t1[1] + traction_t2 * t2[1],
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traction_n * n[2] + traction_t1 * t1[2] + traction_t2 * t2[2],
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])
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}
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/// Route A: the surface integral of the reconstructed traction over the
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/// body's samples at spacing `ds`.
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pub fn surface_force(&self, body: &Body, f: &Field, mu: f64, t: f64, ds: f64) -> SurfaceForce {
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let samples = body.surface_samples(ds);
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let mut force = [0.0; 3];
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let mut skipped = 0;
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for s in &samples {
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match self.traction_at(body, f, mu, t, [s.x, s.y, s.z], [s.nx, s.ny, s.nz]) {
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Some(tr) => {
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for c in 0..3 {
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force[c] += tr[c] * s.area;
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}
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}
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None => skipped += 1,
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}
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}
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SurfaceForce {
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f: force,
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samples: samples.len(),
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skipped,
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}
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}
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/// Route B: the momentum balance over the box of whole cells
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/// `[i0, i1) × [j0, j1) × [k0, k1)` (in the fluid on its boundary):
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/// `F = Σ_outer (σ·n − ρ u (u·n)) A − d/dt ∫ ρ u dV + ∫ f dV`.
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#[allow(clippy::too_many_arguments)]
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pub fn control_volume_force(
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&self,
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f: &Field,
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dt: f64,
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rho: f64,
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mu: f64,
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source: Option<&dyn Fn(f64, f64, f64) -> (f64, f64, f64)>,
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bx: (usize, usize, usize, usize, usize, usize),
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) -> [f64; 3] {
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self.control_volume_force_with_walls(f, dt, rho, mu, source, bx, false)
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}
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/// Route B on a box spanning the whole z range between NO-SLIP z walls
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/// (`no_slip_z`): the walls' shear on the fluid inside the box (against
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/// a wall at rest) is an outer-face stress and enters the balance;
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/// without it (slip or periodic sides) the z faces carry nothing.
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#[allow(clippy::too_many_arguments)]
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pub fn control_volume_force_with_walls(
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&self,
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f: &Field,
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dt: f64,
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rho: f64,
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mu: f64,
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source: Option<&dyn Fn(f64, f64, f64) -> (f64, f64, f64)>,
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(i0, i1, j0, j1, k0, k1): (usize, usize, usize, usize, usize, usize),
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no_slip_z: bool,
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) -> [f64; 3] {
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let g = self.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 (u, v, w, p) = (&f.u, &f.v, &f.w, &f.p);
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let mut force = [0.0; 3];
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let uc =
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|k: usize, j: usize, i: usize| 0.5 * (u[g.uface(k, j, i)] + u[g.uface(k, j, i + 1)]);
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let vc =
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|k: usize, j: usize, i: usize| 0.5 * (v[g.vface(k, j, i)] + v[g.vface(k, j + 1, i)]);
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let wc =
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|k: usize, j: usize, i: usize| 0.5 * (w[g.wface(k, j, i)] + w[g.wface(k + 1, j, i)]);
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// Central inside, one-sided at the domain edge.
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let dd = |prev: Option<f64>, here: f64, next: Option<f64>, h: f64| match (prev, next) {
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(Some(a), Some(b)) => (b - a) / (2.0 * h),
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(None, Some(b)) => (b - here) / h,
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(Some(a), None) => (here - a) / h,
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(None, None) => 0.0,
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};
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// x faces: u lives there.
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for k in k0..k1 {
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for j in j0..j1 {
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for (i, sign) in [(i1, 1.0), (i0, -1.0)] {
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let un = u[g.uface(k, j, i)];
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let p_f = 0.5 * (p[g.cell(k, j, i - 1)] + p[g.cell(k, j, i)]);
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let dudx = (u[g.uface(k, j, i + 1)] - u[g.uface(k, j, i - 1)]) / (2.0 * dx);
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let dvdx = (vc(k, j, i) - vc(k, j, i - 1)) / dx;
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let dwdx = (wc(k, j, i) - wc(k, j, i - 1)) / dx;
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let dudy = dd(
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(j > 0).then(|| u[g.uface(k, j - 1, i)]),
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un,
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(j + 1 < ny).then(|| u[g.uface(k, j + 1, i)]),
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dy,
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);
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let dudz = dd(
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(k > 0).then(|| u[g.uface(k - 1, j, i)]),
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un,
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(k + 1 < nz).then(|| u[g.uface(k + 1, j, i)]),
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dz,
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);
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let v_f = 0.5 * (vc(k, j, i - 1) + vc(k, j, i));
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let w_f = 0.5 * (wc(k, j, i - 1) + wc(k, j, i));
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let a = dy * dz;
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force[0] += sign * ((-p_f + 2.0 * mu * dudx) - rho * un * un) * a;
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force[1] += sign * (mu * (dudy + dvdx) - rho * v_f * un) * a;
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force[2] += sign * (mu * (dudz + dwdx) - rho * w_f * un) * a;
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}
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}
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}
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// y faces: v lives there.
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for k in k0..k1 {
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for i in i0..i1 {
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for (j, sign) in [(j1, 1.0), (j0, -1.0)] {
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let vn = v[g.vface(k, j, i)];
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let p_f = 0.5 * (p[g.cell(k, j - 1, i)] + p[g.cell(k, j, i)]);
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let dvdy = (v[g.vface(k, j + 1, i)] - v[g.vface(k, j - 1, i)]) / (2.0 * dy);
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let dudy = (uc(k, j, i) - uc(k, j - 1, i)) / dy;
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let dwdy = (wc(k, j, i) - wc(k, j - 1, i)) / dy;
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let dvdx = dd(
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(i > 0).then(|| v[g.vface(k, j, i - 1)]),
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vn,
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(i + 1 < nx).then(|| v[g.vface(k, j, i + 1)]),
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dx,
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);
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let dvdz = dd(
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(k > 0).then(|| v[g.vface(k - 1, j, i)]),
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vn,
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(k + 1 < nz).then(|| v[g.vface(k + 1, j, i)]),
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dz,
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);
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let u_f = 0.5 * (uc(k, j - 1, i) + uc(k, j, i));
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let w_f = 0.5 * (wc(k, j - 1, i) + wc(k, j, i));
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let a = dx * dz;
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force[0] += sign * (mu * (dudy + dvdx) - rho * u_f * vn) * a;
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force[1] += sign * ((-p_f + 2.0 * mu * dvdy) - rho * vn * vn) * a;
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force[2] += sign * (mu * (dvdz + dwdy) - rho * w_f * vn) * a;
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}
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}
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}
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// 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;
|
||
if full_span && no_slip_z {
|
||
let a = dx * dy;
|
||
for j in j0..j1 {
|
||
for i in i0..i1 {
|
||
for k in [0, nz - 1] {
|
||
if !self.is_fluid_cell(g.cell(k, j, i)) {
|
||
continue;
|
||
}
|
||
force[0] -= mu * uc(k, j, i) / (0.5 * dz) * a;
|
||
force[1] -= mu * vc(k, j, i) / (0.5 * dz) * a;
|
||
}
|
||
}
|
||
}
|
||
}
|
||
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 dv = dv * self.vol(idx);
|
||
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
|
||
}
|
||
}
|