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empirical.rs
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/
empirical.rs
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use cgmath::{self, InnerSpace, Vector3};
use fft;
use math::{integration, Real};
use ndarray::{Array2, ArrayView2, ArrayViewMut2, Axis};
use num::Zero;
use num::complex::Complex;
use rand;
use rand::distributions::normal;
use std::f32::consts::PI;
use std::sync::Arc;
fn dispersion_peak<T: Real>(gravity: T, wind_speed: T, fetch: T) -> T {
// Note: pow(x, 1/3) is missing in [Horvath2015]
T::new(22.0) * (gravity.powi(2) / (wind_speed * fetch)).powf(T::new(1.0/3.0))
}
/// Representing a spectral density function of angular frequency.
///
/// This is only the non-directional component of the spectrum following [Horvath15].
pub trait Spectrum<T: Real>: Sync {
fn evaluate(&self, omega: T) -> T;
}
/// Joint North Sea Wave Observation Project (JONSWAP) Spectrum [Horvath15] Section 5.1.4
pub struct SpectrumJONSWAP<T: Real> {
pub wind_speed: T, // [m/s]
pub fetch: T,
pub gravity: T, // [m/s^2]
}
impl<T: Real> Spectrum<T> for SpectrumJONSWAP<T> {
// [Horvath15] Eq. 28
fn evaluate(&self, omega: T) -> T {
if omega < T::default_epsilon() {
return T::zero();
}
let gamma = T::new(3.3);
let omega_peak = dispersion_peak(self.gravity, self.wind_speed, self.fetch);
let alpha =
T::new(0.076) * (self.wind_speed.powi(2) / (self.fetch * self.gravity)).powf(T::new(0.22));
let sigma = T::new(if omega <= omega_peak { 0.07 } else { 0.09 });
let r = (-(omega - omega_peak).powi(2) / (T::new(2.0) * (sigma * omega_peak).powi(2))).exp();
(alpha * (self.gravity).powi(2) / omega.powi(5)) * (T::new(-5.0/4.0) * (omega_peak/omega).powi(4)).exp() * gamma.powf(r)
}
}
/// Texel MARSEN ARSLOE (TMA) Spectrum [Horvath15] Section 5.1.5
pub struct SpectrumTMA<T: Real> {
pub jonswap: SpectrumJONSWAP<T>,
pub depth: T, // TODO: [m]
}
impl<T: Real> SpectrumTMA<T> {
/// Kitaigorodskii Depth Attenuation Function [Horvath15] Eq. 29
///
/// Using the approximation from Thompson and Vincent, 1983
/// as proposed in Section 5.1.5.
fn kitaigorodskii_depth_attenuation(&self, omega: T) -> T {
let omega_h = (omega * (self.depth / self.jonswap.gravity)).max(T::zero()).min(T::new(2.0));
if omega_h <= T::one() {
T::new(0.5) * omega_h.powi(2)
} else {
T::one() - T::new(0.5) * (T::new(2.0) - omega_h).powi(2)
}
}
}
impl<T: Real> Spectrum<T> for SpectrumTMA<T> {
fn evaluate(&self, omega: T) -> T {
self.jonswap.evaluate(omega) * self.kitaigorodskii_depth_attenuation(omega)
}
}
pub struct Parameters<T> {
pub surface_tension: T,
pub water_density: T,
pub water_depth: T,
pub gravity: T, // [m/s^2]
pub wind_speed: T, // [m/s]
pub fetch: T,
pub swell: T,
pub domain_size: T,
}
fn sample_spectrum<S, T>(
parameters: &Parameters<T>,
spectrum: &S,
pos: cgmath::Vector2<T>,
) -> (Complex<T>, T)
where
S: Spectrum<T>,
T: Real,
{
if pos.magnitude() < T::default_epsilon() {
return (Complex::new(T::zero(), T::zero()), T::zero());
}
let theta = (pos.y).atan2(pos.x);
let grad_k = T::new(2.0 * PI) / parameters.domain_size;
let (omega, grad_omega) = dispersion_capillary(parameters, pos.magnitude());
let spreading = directional_spreading(parameters, omega, theta, directional_base_donelan_banner);
let sample = spectrum.evaluate(omega);
let normal::StandardNormal(z) = rand::random();
let phase = T::new(2.0 * PI) * rand::random::<T>();
let amplitude = T::new(z as f32) * (T::new(2.0) * spreading * sample * grad_k.powi(2) * grad_omega / pos.magnitude()).sqrt();
(Complex::new(phase.cos() * amplitude, phase.sin() * amplitude), omega)
}
fn dispersion_capillary<T>(parameters: &Parameters<T>, wave_number: T) -> (T, T)
where
T: Real,
{
let sech = |x: T| { T::one() / x.cosh() };
let sigma = parameters.surface_tension;
let rho = parameters.water_density;
let g = parameters.gravity;
let h = parameters.water_depth;
let k = wave_number;
let dispersion = ((g*k + (sigma/rho) * k.powi(3)) * (h*k).tanh()).sqrt();
let grad_dispersion = (
h * sech(h*k).powi(2) * (g*k + (sigma/rho) * k.powi(3)) +
(h*k).tanh() * (g + T::new(3.0)*(sigma/rho) * k.powi(2))
) / (T::new(2.0) * dispersion);
(dispersion, grad_dispersion)
}
// [Horvath15] Eq. 44
fn directional_elongation<T: Real>(parameters: &Parameters<T>, omega: T, theta: T) -> T {
let shaping = {
let omega_peak = dispersion_peak(parameters.gravity, parameters.wind_speed, parameters.fetch);
T::new(16.0) * (omega_peak / omega).tanh() * parameters.swell.powi(2)
};
(theta/T::new(2.0)).cos().abs().powf(T::new(2.0)*shaping)
}
fn directional_spreading<F, T>(parameters: &Parameters<T>, omega: T, theta: T, directional_base: F) -> T
where
F: Fn(&Parameters<T>, T, T) -> T,
T: Real,
{
let pi = T::new(PI);
let normalization =
integration::trapezoidal_quadrature(
(-pi, pi),
128,
|theta| directional_base(parameters, omega, theta) * directional_elongation(parameters, omega, theta));
directional_base(parameters, omega, theta) * directional_elongation(parameters, omega, theta) / normalization
}
// Donelan-Banner Directional Spreading [Horvath15] Eq. 38
fn directional_base_donelan_banner<T>(parameters: &Parameters<T>, omega: T, theta: T) -> T
where
T: Real,
{
let beta = {
let omega_peak = dispersion_peak(parameters.gravity, parameters.wind_speed, parameters.fetch);
let omega_ratio = omega/omega_peak;
if omega_ratio < T::new(0.95) {
T::new(2.61) * omega_ratio.powf(T::new(1.3))
} else if omega_ratio < T::new(1.6) {
T::new(2.28) * omega_ratio.powf(T::new(-1.3))
} else {
let epsilon = T::new(-0.4) + T::new(0.8393) * (T::new(-0.567) * (omega_ratio.powi(2)).ln()).exp();
T::new(10).powf(epsilon)
}
};
let sech = |x: T| { T::one() / x.cosh() };
beta / (T::new(2.0) * (beta * T::new(PI)).tanh()) * sech(beta * theta).powi(2)
}
pub struct Ocean<T> {
resolution: usize,
fft_plan: fft::FFTplanner<T>,
fft_buffer: Array2<Complex<T>>,
displacement_x: Array2<Complex<T>>,
displacement_y: Array2<Complex<T>>,
displacement_z: Array2<Complex<T>>,
}
impl<T> Ocean<T>
where
T: Real + fft::FFTnum,
{
pub fn new(resolution: usize) -> Self {
Ocean {
fft_plan: fft::FFTplanner::new(true),
fft_buffer: Array2::from_elem((resolution, resolution), Complex::new(T::zero(), T::zero())),
resolution,
displacement_x: Self::new_map(resolution),
displacement_y: Self::new_map(resolution),
displacement_z: Self::new_map(resolution),
}
}
pub fn build_height_spectrum<S>(
&self,
parameters: &Parameters<T>,
spectrum: &S,
) -> (Array2<Complex<T>>, Array2<T>)
where
S: Spectrum<T>,
{
let resolution = self.resolution;
let pi = T::new(PI);
let mut height_spectrum = Array2::from_elem((resolution, resolution), Complex::new(T::zero(), T::zero()));
let mut omega = Array2::zeros((resolution, resolution));
par_azip!(
index (j, i),
mut height_spectrum,
mut omega,
in {
let x = T::new(2 * i as isize - resolution as isize - 1);
let y = T::new(2 * j as isize - resolution as isize - 1);
let sample = {
let k = cgmath::vec2(
pi * x / parameters.domain_size,
pi * y / parameters.domain_size,
);
sample_spectrum(parameters, spectrum, k)
};
*height_spectrum = sample.0;
*omega = sample.1;
});
(height_spectrum, omega)
}
fn new_map(resolution: usize) -> Array2<Complex<T>> {
Array2::from_elem((resolution, resolution), Complex::new(T::zero(), T::zero()))
}
pub fn new_displacement(&self) -> Array2<Vector3<T>> {
Array2::from_elem((self.resolution, self.resolution), Vector3::zero())
}
pub fn propagate(
&mut self,
time: T,
parameters: &Parameters<T>,
samples: ArrayView2<Complex<T>>,
omega: ArrayView2<T>,
mut displacement: ArrayViewMut2<Vector3<T>>)
{
let resolution = self.resolution;
let pi = T::new(PI);
// propgation step
par_azip!(
index (j, i),
omega,
ref dx (&mut self.displacement_x),
ref dy (&mut self.displacement_y),
ref dz (&mut self.displacement_z),
in {
let x = T::new(2 * i as isize - resolution as isize - 1);
let y = T::new(2 * j as isize - resolution as isize - 1);
let k = cgmath::vec2(
pi * x / parameters.domain_size,
pi * y / parameters.domain_size,
);
let dispersion = omega * time;
let disp_pos = Complex::new(dispersion.cos(), dispersion.sin());
let disp_neg = Complex::new(dispersion.cos(),-dispersion.sin());
let sample = samples[(j, i)] * disp_pos + samples[(resolution-j-1, resolution-i-1)] * disp_neg;
let k_normalized = {
let len = k.magnitude();
if len < T::default_epsilon() {
Complex::new(T::zero(), T::zero())
} else {
Complex::new(k.x / len, k.y / len)
}
};
*dx = Complex::new(T::zero(), -k_normalized.re) * sample;
*dy = sample;
*dz = Complex::new(T::zero(), -k_normalized.im) * sample;
});
let plan = self.fft_plan.plan_fft(self.resolution);
Self::spectral_to_spatial(&plan, self.displacement_x.view_mut(), self.fft_buffer.view_mut());
// correction step
par_azip!(
index (j, i),
src (&self.fft_buffer),
ref dst (&mut displacement)
in {
if (j+i) % 2 == 0 {
dst.x = -src.re;
} else {
dst.x = src.re;
}
});
Self::spectral_to_spatial(&plan, self.displacement_y.view_mut(), self.fft_buffer.view_mut());
// correction step
par_azip!(
index (j, i),
src (&self.fft_buffer),
ref dst (&mut displacement)
in {
if (j+i) % 2 == 0 {
dst.y = -src.re;
} else {
dst.y = src.re;
}
});
Self::spectral_to_spatial(&plan, self.displacement_z.view_mut(), self.fft_buffer.view_mut());
// correction step
par_azip!(
index (j, i),
src (&self.fft_buffer),
ref dst (&mut displacement)
in {
if (j+i) % 2 == 0 {
dst.z = -src.re;
} else {
dst.z = src.re;
}
});
}
// Transform a spatial 2d field into a spatial field
// Output is stored in self.fft_buffer
fn spectral_to_spatial(plan: &Arc<fft::FFT<T>>, mut input: ArrayViewMut2<Complex<T>>, mut output: ArrayViewMut2<Complex<T>>) {
par_azip!(
mut src (input.axis_iter_mut(Axis(0)))
mut dst (output.axis_iter_mut(Axis(0)))
in {
plan.process(src.as_slice_mut().unwrap(), dst.as_slice_mut().unwrap());
});
input.assign(&output.t());
par_azip!(
mut src (input.axis_iter_mut(Axis(0)))
mut dst (output.axis_iter_mut(Axis(0)))
in {
plan.process(src.as_slice_mut().unwrap(), dst.as_slice_mut().unwrap());
});
}
}