sva-samples 0.7.15

Buffers, frames, the f64 machine, the collapse of a law onto a grid, and the measured observations over one
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172

// Concern: a pinned stiff string's energy and its felt's press | Non-concern: stepping the grid (stiff_string.rs), the strike | IO: (&StringGrid) -> energy

use crate::physics::stiff_string::StringGrid;

const U: f64 = f64::EPSILON / 2.0;

pub(crate) fn gamma(n: f64) -> f64 {
    n * U / (1.0 - n * U)
}

fn slopes(y: &[f64], n: usize) -> impl Iterator<Item = f64> + '_ {
    (0..n).map(move |j| pinned(y, n, j + 1) - pinned(y, n, j))
}

fn curvatures(y: &[f64], n: usize) -> impl Iterator<Item = f64> + '_ {
    (1..n).map(move |j| pinned(y, n, j + 1) - 2.0 * y[j] + pinned(y, n, j - 1))
}

/// Node `j`, the agraffe pin at 0 read as zero; `y[n]` holds a rigid pin's zero or the bridge.
fn pinned(y: &[f64], _n: usize, j: usize) -> f64 {
    match j {
        0 => 0.0,
        j => y[j],
    }
}

/// `(j, k psi dt^2/(rho dx), R psi dt/(2 rho dx))`, `psi` node `j`'s share of the felt.
pub(crate) type Felt = (usize, f64, f64);

/// Spring and dashpot centred on `y^n`.
pub(crate) fn press(grid: &mut StringGrid, felt: &[Felt]) {
    for &(j, kappa, rho) in felt {
        grid.y_next[j] =
            (grid.y_next[j] + (rho - kappa / 2.0) * grid.y_prev[j]) / (1.0 + kappa / 2.0 + rho);
    }
}

/// `E = w (v'(I - A/2)v + y'K y- + (y'ky + y-'ky-)/2 + lambda^2 v_n^2/2)/2` joules, `k` the
/// felt's springs, `v_n` the bridge's step, and a bound over its rounding: each sum errs by at
/// most `gamma` of it taken over magnitudes.
pub(crate) fn energy(grid: &StringGrid, dt: f64, felt: &[Felt]) -> (f64, f64) {
    let (n, y, yp) = (grid.n, &grid.y_now, &grid.y_prev);
    let v: Vec<f64> = y.iter().zip(yp).map(|(a, b)| a - b).collect();
    let size: Vec<f64> = y.iter().zip(yp).map(|(a, b)| a.abs() + b.abs()).collect();
    let (y_abs, yp_abs): (Vec<f64>, Vec<f64>) =
        y.iter().zip(yp).map(|(a, b)| (a.abs(), b.abs())).unzip();
    let squares = v[1..n].iter().map(|x| x * x).sum::<f64>();
    let bent = slopes(&v, n).map(|x| x * x).sum::<f64>();
    let tension = slopes(y, n)
        .zip(slopes(yp, n))
        .map(|(a, b)| a * b)
        .sum::<f64>();
    let bending = curvatures(y, n)
        .zip(curvatures(yp, n))
        .map(|(a, b)| a * b)
        .sum::<f64>();
    let spring: f64 = felt
        .iter()
        .map(|&(j, kappa, _)| kappa * (y[j] * y[j] + yp[j] * yp[j]) / 2.0)
        .sum();
    let value = (1.0 - grid.damp_a / 2.0) * squares - grid.damp_b / 2.0 * bent
        + grid.courant_sq * tension
        + grid.stiff_sq * bending
        + spring
        + grid.courant_sq / 2.0 * v[n] * v[n];
    let spread = (1.0 + grid.damp_a / 2.0) * size[1..n].iter().map(|x| x * x).sum::<f64>()
        + grid.damp_b / 2.0 * sums(&size, n).map(|x| x * x).sum::<f64>()
        + grid.courant_sq
            * sums(&y_abs, n)
                .zip(sums(&yp_abs, n))
                .map(|(a, b)| a * b)
                .sum::<f64>()
        + grid.stiff_sq
            * bends(&y_abs, n)
                .zip(bends(&yp_abs, n))
                .map(|(a, b)| a * b)
                .sum::<f64>()
        + spring
        + grid.courant_sq / 2.0 * size[n] * size[n];
    let w = grid.rho * grid.dx / (dt * dt) / 2.0;
    let joules = w * value;
    let slack = w * spread * gamma(2.0 * n as f64 + 96.0);
    (joules, (joules + slack) * (1.0 + 4.0 * U))
}

/// [`slopes`] and [`curvatures`] over magnitudes, every difference taken as a sum.
fn sums(m: &[f64], n: usize) -> impl Iterator<Item = f64> + '_ {
    (0..n).map(move |j| pinned(m, n, j + 1) + pinned(m, n, j))
}

fn bends(m: &[f64], n: usize) -> impl Iterator<Item = f64> + '_ {
    (1..n).map(move |j| pinned(m, n, j + 1) + 2.0 * m[j] + pinned(m, n, j - 1))
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::physics::twofold::Twofold;

    /// A stiff string bent smoothly and far, so each curvature cancels all but a sliver.
    fn bent_far(n: usize) -> StringGrid {
        let shape = |j: usize, phase: f64| {
            let x = std::f64::consts::PI * j as f64 / n as f64;
            1e3 * (x.sin() * phase.cos() + (3.0 * x).sin() * phase.sin() / 7.0)
        };
        let mut y_now: Vec<f64> = (0..=n).map(|j| shape(j, 0.3)).collect();
        let mut y_prev: Vec<f64> = (0..=n).map(|j| shape(j, 0.31)).collect();
        for y in [&mut y_now, &mut y_prev] {
            y[0] = 0.0;
            y[n] = 0.0;
        }
        StringGrid {
            y_next: vec![0.0; n + 1],
            y_now,
            y_prev,
            n,
            dx: 0.01,
            rho: 8.9e-3,
            courant_sq: 1e-4,
            stiff_sq: 0.24,
            damp_a: 1e-5,
            damp_b: 1e-4,
        }
    }

    fn exact(grid: &StringGrid, dt: f64, felt: &[Felt]) -> f64 {
        let t = Twofold::of;
        let (n, y, yp) = (grid.n, &grid.y_now, &grid.y_prev);
        let at = |z: &[f64], j: usize| t(pinned(z, n, j));
        let v = |j: usize| at(y, j).sub(at(yp, j));
        let bend = |z: &[f64], j: usize| at(z, j + 1).sub(t(2.0).mul(t(z[j]))).add(at(z, j - 1));
        let mut value = t(0.0);
        for j in 1..n {
            let own = t(1.0).sub(t(grid.damp_a).div(t(2.0)));
            value = value.add(own.mul(v(j)).mul(v(j)));
            let curved = bend(y, j).mul(bend(yp, j));
            value = value.add(t(grid.stiff_sq).mul(curved));
        }
        for j in 0..n {
            let dv = v(j + 1).sub(v(j));
            value = value.sub(t(grid.damp_b).div(t(2.0)).mul(dv).mul(dv));
            let slope = at(y, j + 1).sub(at(y, j)).mul(at(yp, j + 1).sub(at(yp, j)));
            value = value.add(t(grid.courant_sq).mul(slope));
        }
        for &(j, kappa, _) in felt {
            let pair = t(y[j]).mul(t(y[j])).add(t(yp[j]).mul(t(yp[j])));
            value = value.add(t(kappa).mul(pair).div(t(2.0)));
        }
        let w = t(grid.rho)
            .mul(t(grid.dx))
            .div(t(dt).mul(t(dt)))
            .div(t(2.0));
        w.mul(value).value()
    }

    #[test]
    fn a_string_energy_bound_covers_its_rounding_where_every_curvature_cancels() {
        let dt = 1.0 / 44_100.0;
        for n in [40, 400, 1600] {
            let grid = bent_far(n);
            let felt = [(n / 7, 3e-3, 0.0), (n / 7 + 1, 3e-3, 0.0)];
            for felt in [&[][..], &felt[..]] {
                let (joules, upper) = energy(&grid, dt, felt);
                let truth = exact(&grid, dt, felt);
                assert!(
                    (truth - joules).abs() <= upper - joules,
                    "n {n}: E {truth}, computed {joules}, bound {upper}"
                );
            }
        }
    }
}