cellular_raza_building_blocks/cell_building_blocks/mechanics.rs
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use cellular_raza_concepts::{CalcError, Mechanics, RngError};
use itertools::Itertools;
use nalgebra::{SMatrix, SVector};
use serde::{Deserialize, Serialize};
#[cfg(feature = "pyo3")]
use pyo3::prelude::*;
macro_rules! implement_newton_damped_mechanics(
(
$struct_name:ident,
$d:literal
) => {
implement_newton_damped_mechanics!($struct_name, $d, f64);
};
(
$struct_name:ident,
$d:literal,
$float_type:ty
) => {
/// Newtonian dynamics governed by mass and damping.
///
/// # Parameters
/// | Symbol | Parameter | Description |
/// | --- | --- | --- |
/// | $\vec{x}$ | `pos` | Position of the particle. |
/// | $\dot{\vec{x}}$ | `vel` | Velocity of the particle. |
/// | $\lambda$ | `damping` | Damping constant |
/// | $m$ | `mass` | Mass of the particle. |
///
/// # Equations
/// The equation of motion is given by
/// \\begin{equation}
/// m \ddot{\vec{x}} = \vec{F} - \lambda \dot{\vec{x}}
/// \\end{equation}
/// where $\vec{F}$ is the force as calculated by the
/// [Interaction](cellular_raza_concepts::Interaction) trait.
///
/// # Comments
/// If the cell is growing, we need to increase the mass $m$.
/// By interacting with the outside world, we can adapt $\lambda$ to external values
/// although this is rarely desirable.
/// Both operations need to be implemented by other concepts such as
/// [Cycle](cellular_raza_concepts::Cycle).
#[derive(Clone, Debug, Serialize, Deserialize, PartialEq)]
#[cfg_attr(feature = "pyo3", pyclass)]
pub struct $struct_name {
/// Current position $\vec{x}$ given by a vector of dimension `D`.
pub pos: SVector<$float_type, $d>,
/// Current velocity $\dot{\vec{x}}$ given by a vector of dimension `D`.
pub vel: SVector<$float_type, $d>,
/// Damping constant $\lambda$.
pub damping_constant: $float_type,
/// Mass $m$ of the object.
pub mass: $float_type,
}
impl $struct_name {
#[doc = "Create a new "]
#[doc = stringify!($struct_name)]
/// from position, velocity, damping constant and mass
pub fn new(
pos: [$float_type; $d],
vel: [$float_type; $d],
damping_constant: $float_type,
mass: $float_type,
) -> Self {
Self {
pos: pos.into(),
vel: vel.into(),
damping_constant,
mass,
}
}
}
#[cfg(feature = "pyo3")]
#[pymethods]
#[cfg_attr(docsrs, doc(cfg(feature = "pyo3")))]
impl $struct_name {
#[new]
fn _new(
pos: [$float_type; $d],
vel: [$float_type; $d],
damping_constant: $float_type,
mass: $float_type,
) -> Self {
Self::new(pos, vel, damping_constant, mass)
}
/// [pyo3] getter for `pos`
#[getter]
pub fn get_pos(&self) -> [$float_type; $d] {
self.pos.into()
}
/// [pyo3] getter for `vel`
#[getter]
pub fn get_vel(&self) -> [$float_type; $d] {
self.vel.into()
}
/// [pyo3] getter for `damping_constant`
#[getter]
pub fn get_damping_constant(&self) -> $float_type {
self.damping_constant
}
/// [pyo3] getter for `mass`
#[getter]
pub fn get_mass(&self) -> $float_type {
self.mass
}
/// [pyo3] setter for `pos`
#[setter]
pub fn set_pos(&mut self, pos: [$float_type; $d]) {
self.pos = pos.into();
}
/// [pyo3] setter for `vel`
#[setter]
pub fn set_vel(&mut self, vel: [$float_type; $d]) {
self.vel = vel.into();
}
/// [pyo3] setter for `damping_constant`
#[setter]
pub fn set_damping_constant(&mut self, damping_constant: $float_type) {
self.damping_constant = damping_constant;
}
/// [pyo3] setter for `mass`
#[setter]
pub fn set_mass(&mut self, mass: $float_type) {
self.mass = mass;
}
}
impl Mechanics<
SVector<$float_type, $d>,
SVector<$float_type, $d>,
SVector<$float_type, $d>,
$float_type
> for $struct_name
{
fn get_random_contribution(
&self,
_: &mut rand_chacha::ChaCha8Rng,
_dt: $float_type,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), RngError> {
Ok((num::Zero::zero(), num::Zero::zero()))
}
fn calculate_increment(
&self,
force: SVector<$float_type, $d>,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), CalcError> {
let dx = self.vel;
let dv = force / self.mass - self.damping_constant * self.vel;
Ok((dx, dv))
}
}
impl cellular_raza_concepts::Position<SVector<$float_type, $d>> for $struct_name {
fn pos(&self) -> SVector<$float_type, $d> {
self.pos
}
fn set_pos(&mut self, pos: &SVector<$float_type, $d>) {
self.pos = *pos;
}
}
impl cellular_raza_concepts::Velocity<SVector<$float_type, $d>> for $struct_name {
fn velocity(&self) -> SVector<$float_type, $d> {
self.vel
}
fn set_velocity(&mut self, velocity: &SVector<$float_type, $d>) {
self.vel = *velocity;
}
}
}
);
implement_newton_damped_mechanics!(NewtonDamped1D, 1);
implement_newton_damped_mechanics!(NewtonDamped2D, 2);
implement_newton_damped_mechanics!(NewtonDamped3D, 3);
implement_newton_damped_mechanics!(NewtonDamped1DF32, 1, f32);
implement_newton_damped_mechanics!(NewtonDamped2DF32, 2, f32);
implement_newton_damped_mechanics!(NewtonDamped3DF32, 3, f32);
/// Generate a vector corresponding to a wiener process.
///
/// This function calculates a statically sized random vector with dimension `D`.
/// It uses a [rand_distr::StandardNormal] distribution and divides the result by `dt` such that
/// the correct incremental wiener process is obtained.
pub fn wiener_process<F, const D: usize>(
rng: &mut rand_chacha::ChaCha8Rng,
dt: F,
) -> Result<SVector<F, D>, RngError>
where
F: core::ops::DivAssign + nalgebra::Scalar + num::Float,
rand_distr::StandardNormal: rand_distr::Distribution<F>,
{
let std_dev = dt.sqrt();
let distr = match rand_distr::Normal::new(F::zero(), std_dev) {
Ok(e) => Ok(e),
Err(e) => Err(RngError(format!("{e}"))),
}?;
let random_dir = SVector::<F, D>::from_distribution(&distr, rng);
Ok(random_dir / dt)
}
macro_rules! implement_brownian_mechanics(
($struct_name:ident, $d:literal, $float_type:ty) => {
/// Brownian motion of particles
///
/// # Parameters & Variables
/// | Symbol | Struct Field | Description |
/// | --- | --- | --- |
/// | $D$ | `diffusion_constant` | Damping constant of each particle. |
/// | $k_BT$ | `kb_temperature` | Product of temperature $T$ and Boltzmann constant $k_B$. |
/// | | | |
/// | $\vec{x}$ | `pos` | Position of the particle. |
/// | $R(t)$ | (automatically generated) | Gaussian process |
///
/// # Equations
/// We integrate the standard brownian motion stochastic differential equation.
/// \\begin{equation}
/// \dot{\vec{x}} = -\frac{D}{k_B T}\nabla V(x) + \sqrt{2D}R(t)
/// \\end{equation}
/// The new random vector is then also sampled by a distribution with greater width.
/// If we choose this value larger than one, we can
/// resolve smaller timesteps to more accurately solve the equations.
#[derive(Clone, Debug, Deserialize, Serialize, PartialEq)]
#[cfg_attr(feature = "pyo3", pyclass)]
pub struct $struct_name {
/// Current position of the particle $\vec{x}$.
pub pos: SVector<$float_type, $d>,
/// Diffusion constant $D$.
pub diffusion_constant: $float_type,
/// The product of temperature and boltzmann constant $k_B T$.
pub kb_temperature: $float_type,
}
impl $struct_name {
/// Constructs a new
#[doc = concat!("[", stringify!($struct_name), "]")]
pub fn new(
pos: [$float_type; $d],
diffusion_constant: $float_type,
kb_temperature: $float_type,
) -> Self {
Self {
pos: pos.into(),
diffusion_constant,
kb_temperature,
}
}
}
#[cfg(feature = "pyo3")]
#[pymethods]
#[cfg_attr(docsrs, doc(cfg(feature = "pyo3")))]
impl $struct_name {
#[new]
fn _new(
pos: [$float_type; $d],
diffusion_constant: $float_type,
kb_temperature: $float_type,
) -> Self {
Self::new(pos, diffusion_constant, kb_temperature)
}
/// [pyo3] setter for `pos`
#[setter]
pub fn set_pos(&mut self, pos: [$float_type; $d]) {
self.pos = pos.into();
}
/// [pyo3] setter for `diffusion_constant`
#[setter]
pub fn set_diffusion_constant(&mut self, diffusion_constant: $float_type) {
self.diffusion_constant = diffusion_constant;
}
/// [pyo3] setter for `kb_temperature`
#[setter]
pub fn set_kb_temperature(&mut self, kb_temperature: $float_type) {
self.kb_temperature = kb_temperature;
}
/// [pyo3] getter for `pos`
#[getter]
pub fn get_pos(&self) -> [$float_type; $d] {
self.pos.into()
}
/// [pyo3] getter for `diffusion_constant`
#[getter]
pub fn get_diffusion_constant(&self) -> $float_type {
self.diffusion_constant
}
/// [pyo3] getter for `kb_temperature`
#[getter]
pub fn get_kb_temperature(&self) -> $float_type {
self.kb_temperature
}
}
impl Mechanics<
SVector<$float_type, $d>,
SVector<$float_type, $d>,
SVector<$float_type, $d>,
$float_type
> for $struct_name {
fn get_random_contribution(
&self,
rng: &mut rand_chacha::ChaCha8Rng,
dt: $float_type,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), RngError> {
let dpos = (2.0 as $float_type * self.diffusion_constant).sqrt()
* wiener_process(
rng,
dt
)?;
let dvel = SVector::<$float_type, $d>::zeros();
Ok((dpos, dvel))
}
fn calculate_increment(
&self,
force: SVector<$float_type, $d>,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), CalcError> {
use num::Zero;
let dx = self.diffusion_constant / self.kb_temperature * force;
Ok((dx, SVector::<$float_type, $d>::zero()))
}
}
impl cellular_raza_concepts::Position<SVector<$float_type, $d>> for $struct_name {
fn pos(&self) -> SVector<$float_type, $d> {
self.pos
}
fn set_pos(&mut self, pos: &SVector<$float_type, $d>) {
self.pos = *pos;
}
}
impl cellular_raza_concepts::Velocity<SVector<$float_type, $d>> for $struct_name {
fn velocity(&self) -> SVector<$float_type, $d> {
use num::Zero;
SVector::<$float_type, $d>::zero()
}
fn set_velocity(&mut self, _velocity: &SVector<$float_type, $d>) {}
}
}
);
implement_brownian_mechanics!(Brownian1D, 1, f64);
implement_brownian_mechanics!(Brownian2D, 2, f64);
implement_brownian_mechanics!(Brownian3D, 3, f64);
implement_brownian_mechanics!(Brownian1DF32, 1, f32);
implement_brownian_mechanics!(Brownian2DF32, 2, f32);
implement_brownian_mechanics!(Brownian3DF32, 3, f32);
macro_rules! define_langevin_nd(
($struct_name:ident, $d:literal, $float_type:ident) => {
/// Langevin dynamics
///
/// # Parameters & Variables
/// | Symbol | Struct Field | Description |
/// |:---:| --- | --- |
/// | $M$ | `mass` | Mass of the particle. |
/// | $\gamma$ | `damping` | Damping constant |
/// | $k_BT$ | `kb_temperature` | Product of temperature $T$ and Boltzmann constant $k_B$. |
/// | | | |
/// | $\vec{X}$ | `pos` | Position of the particle. |
/// | $\dot{\vec{X}}$ | `vel` | Velocity of the particle. |
/// | $R(t)$ | (automatically generated) | Gaussian process |
///
/// # Equations
///
/// \\begin{equation}
/// M \ddot{\mathbf{X}} = - \mathbf{\nabla} U(\mathbf{X}) - \gamma M\dot{\mathbf{X}} + \sqrt{2 M \gamma k_{\rm B} T}\mathbf{R}(t)
/// \\end{equation}
#[cfg_attr(feature = "pyo3", pyclass)]
#[derive(Clone, Debug, Deserialize, Serialize, PartialEq)]
pub struct $struct_name {
/// Current position
pub pos: SVector<$float_type, $d>,
/// Current velocity
pub vel: SVector<$float_type, $d>,
/// Mass of the object
pub mass: $float_type,
/// Damping constant
pub damping: $float_type,
/// Product of Boltzmann constant and temperature
pub kb_temperature: $float_type,
}
impl Mechanics<
SVector<$float_type, $d>,
SVector<$float_type, $d>,
SVector<$float_type, $d>,
$float_type
> for $struct_name {
fn get_random_contribution(
&self,
rng: &mut rand_chacha::ChaCha8Rng,
dt: $float_type,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), RngError> {
let dvel = (
2.0 as $float_type
* self.damping
* self.kb_temperature
/ self.mass
).sqrt() * wiener_process(
rng,
dt
)?;
let dpos = SVector::<$float_type, $d>::zeros();
Ok((dpos, dvel))
}
fn calculate_increment(
&self,
force: SVector<$float_type, $d>,
) -> Result<(SVector<$float_type, $d>, SVector<$float_type, $d>), CalcError> {
let dx = self.vel;
let dv1 =
1.0 as $float_type / self.mass * force;
let dv2 =
- self.damping * self.vel;
let dv = dv1 + dv2;
Ok((dx, dv))
}
}
impl cellular_raza_concepts::Position<SVector<$float_type, $d>> for $struct_name {
fn pos(&self) -> SVector<$float_type, $d> {
self.pos
}
fn set_pos(&mut self, pos: &SVector<$float_type, $d>) {
self.pos = *pos;
}
}
impl cellular_raza_concepts::Velocity<SVector<$float_type, $d>> for $struct_name {
fn velocity(&self) -> SVector<$float_type, $d> {
self.vel
}
fn set_velocity(&mut self, velocity: &SVector<$float_type, $d>) {
self.vel = *velocity;
}
}
#[cfg(feature = "pyo3")]
#[pymethods]
#[cfg_attr(docsrs, doc(cfg(feature = "pyo3")))]
impl $struct_name {
/// Creates a new [
#[doc = stringify!($struct_name)]
/// ] struct from position, velocity, mass, damping,
/// kb_temperature and the update interval of the mechanics aspect.
#[new]
fn _new(
pos: [$float_type; $d],
vel: [$float_type; $d],
mass: $float_type,
damping: $float_type,
kb_temperature: $float_type,
) -> Self {
Self {
pos: pos.into(),
vel: vel.into(),
mass,
damping,
kb_temperature,
}
}
#[getter(pos)]
/// [pyo3] getter for `position`
pub fn get_position(&self) -> [$float_type; $d] {
self.pos.into()
}
#[setter(pos)]
/// [pyo3] setter for `position`
pub fn set_position(&mut self, pos: [$float_type; $d]) {
self.pos = pos.into();
}
#[getter(damping)]
/// [pyo3] getter for `damping`
pub fn get_damping(&self) -> $float_type {
self.damping
}
#[setter(damping)]
/// [pyo3] setter for `damping`
pub fn set_damping(&mut self, damping: $float_type) {
self.damping = damping;
}
#[getter(mass)]
/// [pyo3] getter for `mass`
pub fn get_mass(&self) -> $float_type {
self.mass
}
#[setter(mass)]
/// [pyo3] setter for `mass`
pub fn set_mass(&mut self, mass: $float_type) {
self.mass = mass;
}
#[getter(kb_temperature)]
/// [pyo3] getter for `kb_temperature`
pub fn get_kb_temperature(&self) -> $float_type {
self.kb_temperature
}
#[setter(kb_temperature)]
/// [pyo3] setter for `kb_temperature`
pub fn set_kb_temperature(&mut self, kb_temperature: $float_type) {
self.kb_temperature = kb_temperature;
}
fn __repr__(&self) -> String {
format!("{self:#?}")
}
}
}
);
define_langevin_nd!(Langevin1D, 1, f64);
define_langevin_nd!(Langevin2D, 2, f64);
define_langevin_nd!(Langevin3D, 3, f64);
define_langevin_nd!(Langevin1DF32, 1, f32);
define_langevin_nd!(Langevin2DF32, 2, f32);
define_langevin_nd!(Langevin3DF32, 3, f32);
/// Mechanics model which represents cells as vertices with edges between them.
///
/// The vertices are attached to each other with springs and a given length between each
/// vertex.
/// Furthermore, we define a central pressure that acts when the total cell area is greater
/// or smaller than the desired one.
/// Each vertex is damped individually by the same constant.
// TODO include more formulas for this model
#[derive(Serialize, Deserialize, Clone, Debug, PartialEq)]
pub struct VertexMechanics2D<const D: usize> {
points: nalgebra::SMatrix<f64, D, 2>,
velocity: nalgebra::SMatrix<f64, D, 2>,
/// Boundary lengths of individual edges
pub cell_boundary_lengths: nalgebra::SVector<f64, D>,
/// Spring tensions of individual edges
pub spring_tensions: nalgebra::SVector<f64, D>,
/// Total cell area
pub cell_area: f64,
/// Central pressure going from middle of the cell outwards
pub central_pressure: f64,
/// Damping constant
pub damping_constant: f64,
/// Controls the random motion of the entire cell
pub diffusion_constant: f64,
}
impl<const D: usize> VertexMechanics2D<D> {
/// Creates a new vertex model in equilibrium conditions.
///
/// The specified parameters are then used to carefully calculate relating properties of the model.
/// We outline the formulas used.
/// Given the number of vertices \\(N\\) in our model (specified by the const generic argument
/// of the [VertexMechanics2D] struct),
/// the resulting average angle when all nodes are equally distributed is
/// \\[
/// \Delta\varphi = \frac{2\pi}{N}
/// \\]
/// Given the total area of the cell ([regular polygon](https://en.wikipedia.org/wiki/Regular_polygon)) \\(A\\), we can calculate the distance from the center point to the individual vertices by inverting
/// \\[
/// A = r^2 \sin\left(\frac{\pi}{N}\right)\cos\left(\frac{\pi}{N}\right).
/// \\]
/// This formula can be derived by elementary geometric arguments.
/// We can then generate the points \\(\vec{p}\_i\\) which make up the cell-boundary by using
/// \\[
/// \vec{p}\_i = \vec{p}\_{mid} + r(\cos(i \Delta\varphi), \sin(i\Delta\varphi))^T.
/// \\]
/// From these points, their distance is calculated and passed as the individual boundary lengths.
/// When randomization is turned on, these points will be slightly randomized in their radius and angle which might lead to non-equilibrium configurations.
/// Pressure, damping and spring tensions are not impacted by randomization.
pub fn new(
middle: SVector<f64, 2>,
cell_area: f64,
rotation_angle: f64,
spring_tensions: f64,
central_pressure: f64,
damping_constant: f64,
diffusion_constant: f64,
randomize: Option<(f64, rand_chacha::ChaCha8Rng)>,
) -> Self {
use rand::Rng;
// Restrict the randomize variable between 0 and 1
let r = match randomize {
Some((rand, _)) => rand.clamp(0.0, 1.0),
_ => 0.0,
};
let rng = || -> f64 {
match randomize {
Some((_, mut rng)) => rng.random_range(0.0..1.0),
None => 0.0,
}
};
// Randomize the overall rotation angle
let rotation_angle = (1.0 - r * rng.clone()()) * rotation_angle;
// Calculate the angle fraction used to determine the points of the polygon
let angle_fraction = std::f64::consts::PI / D as f64;
// Calculate the radius from cell area
let radius = (cell_area / D as f64 / angle_fraction.sin() / angle_fraction.cos()).sqrt();
// TODO this needs to be calculated again
let points = nalgebra::SMatrix::<f64, D, 2>::from_row_iterator((0..D).flat_map(|n| {
let angle =
rotation_angle + 2.0 * angle_fraction * n as f64 * (1.0 - r * rng.clone()());
let radius_modified = radius * (1.0 + 0.5 * r * (1.0 - rng.clone()()));
[
middle.x + radius_modified * angle.cos(),
middle.y + radius_modified * angle.sin(),
]
.into_iter()
}));
// Randomize the boundary lengths
let cell_boundary_lengths = SVector::<f64, D>::from_iterator(
points
.row_iter()
.circular_tuple_windows()
.map(|(p1, p2)| (p1 - p2).norm()),
);
VertexMechanics2D {
points,
velocity: nalgebra::SMatrix::<f64, D, 2>::zeros(),
cell_boundary_lengths,
spring_tensions: SVector::<f64, D>::from_element(spring_tensions),
cell_area,
central_pressure,
damping_constant,
diffusion_constant,
}
}
/// Calculates the boundary length of the regular polygon given the total area in equilibrium.
///
/// The formula used is
/// $$\\begin{align}
/// A &= \frac{L^2}{4n\tan\left(\frac{\pi}{n}\right)}\\\\
/// L &= \sqrt{4An\tan\left(\frac{\pi}{n}\right)}
/// \\end{align}$$
/// where $A$ is the total area, $n$ is the number of vertices and $L$ is the total boundary
/// length.
pub fn calculate_boundary_length(cell_area: f64) -> f64 {
(4.0 * cell_area * (std::f64::consts::PI / D as f64).tan() * D as f64).sqrt()
}
/// Calculates the cell area of the regular polygon in equilibrium.
///
/// The formula used is identical the the one of [Self::calculate_boundary_length].
pub fn calculate_cell_area(boundary_length: f64) -> f64 {
D as f64 * boundary_length.powf(2.0) / (4.0 * (std::f64::consts::PI / D as f64).tan())
}
/// Calculates the current area of the cell
pub fn get_current_cell_area(&self) -> f64 {
0.5_f64
* self
.points
.row_iter()
.circular_tuple_windows()
.map(|(p1, p2)| p1.transpose().perp(&p2.transpose()))
.sum::<f64>()
}
/// Calculate the current polygons boundary length
pub fn calculate_current_boundary_length(&self) -> f64 {
self.points
.row_iter()
.tuple_windows::<(_, _)>()
.map(|(p1, p2)| (p2 - p1).norm())
.sum::<f64>()
}
/// Obtain current cell area
pub fn get_cell_area(&self) -> f64 {
self.cell_area
}
/// Set the current cell area and adjust the length of edges such that the cell is still in
/// equilibrium.
pub fn set_cell_area_and_boundary_length(&mut self, cell_area: f64) {
// Calculate the relative difference to current area
match self.cell_area {
0.0 => {
let new_interaction_parameters = Self::new(
self.points
.row_iter()
.map(|v| v.transpose())
.sum::<nalgebra::Vector2<f64>>(),
cell_area,
0.0,
self.spring_tensions.sum() / self.spring_tensions.len() as f64,
self.central_pressure,
self.damping_constant,
self.diffusion_constant,
None,
);
*self = new_interaction_parameters;
}
_ => {
let relative_length_difference = (cell_area.abs() / self.cell_area.abs()).sqrt();
// Calculate the new length of the cell boundary lengths
self.cell_boundary_lengths
.iter_mut()
.for_each(|length| *length *= relative_length_difference);
self.cell_area = cell_area;
}
};
}
/// Change the internal cell area
pub fn set_cell_area(&mut self, cell_area: f64) {
self.cell_area = cell_area;
}
}
impl VertexMechanics2D<6> {
/// Fills the area of a given rectangle with hexagonal cells. Their orientation is such that
/// the top border has a flat top.
///
/// The produced pattern will like similar to this.
/// ```text
/// __________________________________
/// | ___ ___ ___ |
/// | / \ / \ / \ |
/// | / \___/ \_ ..._/ \ |
/// | \ / \ / \ / |
/// | \___/ \___/ \___/ |
/// | / \ / \ / \ |
/// | . . . . . . |
/// ```
/// The padding around the generated cells will be determined automatically.
pub fn fill_rectangle_flat_top(
cell_area: f64,
spring_tensions: f64,
central_pressure: f64,
damping_constant: f64,
diffusion_constant: f64,
rectangle: [SVector<f64, 2>; 2],
) -> Vec<Self> {
// If the supplied area is larger than the total area, return nothing
let side_x = rectangle[1].x - rectangle[0].x;
let side_y = rectangle[1].y - rectangle[0].y;
if cell_area > side_x * side_y {
return Vec::new();
}
let segment_length = Self::calculate_boundary_length(cell_area) / 6.0;
let radius_outer = Self::outer_radius_from_cell_area(cell_area);
let radius_inner = Self::inner_radius_from_cell_area(cell_area);
// Check if only one single hexagon fits into the domain in any dimension
let n_max_x = (side_x - 2.0 * radius_outer).div_euclid(3.0 / 2.0 * radius_outer) as usize;
let n_max_y = side_y.div_euclid(2.0 * radius_inner);
let total_width_x = 2.0 * radius_outer + (n_max_x - 1) as f64 * 3.0 / 2.0 * radius_outer;
let total_width_y = n_max_y * 2.0 * radius_inner;
let pad_x = (side_x - total_width_x) / 2.0;
let pad_y = (side_y - total_width_y) / 2.0;
let padding = nalgebra::RowVector2::from([pad_x, pad_y]);
let mut generated_models = vec![];
for n_x in 0..n_max_x {
for n_y in 0..n_max_y as usize - n_x % 2 {
let middle = rectangle[0].transpose()
+ padding
+ nalgebra::RowVector2::from([
(1.0 + 3.0 / 2.0 * n_x as f64) * radius_outer,
(1 + 2 * n_y + n_x % 2) as f64 * radius_inner,
]);
let mut pos = nalgebra::SMatrix::<f64, 6, 2>::zeros();
for i in 0..6 {
let phi = 2.0 * std::f64::consts::PI * i as f64 / 6.0;
pos.set_row(
i,
&(middle
+ radius_outer * nalgebra::RowVector2::from([phi.cos(), phi.sin()])),
);
}
generated_models.push(Self {
points: pos,
velocity: SMatrix::zeros(),
cell_boundary_lengths: SVector::from_element(segment_length),
spring_tensions: SVector::from_element(spring_tensions),
cell_area,
central_pressure,
damping_constant,
diffusion_constant,
});
}
}
generated_models
}
}
impl VertexMechanics2D<4> {
/// Fill a specified rectangle with cells of 4 vertices
pub fn fill_rectangle(
cell_area: f64,
spring_tensions: f64,
central_pressure: f64,
damping_constant: f64,
diffusion_constant: f64,
rectangle: [SVector<f64, 2>; 2],
) -> Vec<Self> {
let cell_side_length: f64 = cell_area.sqrt();
let cell_side_length_padded: f64 = cell_side_length * 1.04;
let number_of_cells_x: u64 =
((rectangle[1].x - rectangle[0].x) / cell_side_length_padded).floor() as u64;
let number_of_cells_y: u64 =
((rectangle[1].y - rectangle[0].y) / cell_side_length_padded).floor() as u64;
let start_x: f64 = rectangle[0].x
+ 0.5
* ((rectangle[1].x - rectangle[0].x)
- number_of_cells_x as f64 * cell_side_length_padded);
let start_y: f64 = rectangle[0].y
+ 0.5
* ((rectangle[1].y - rectangle[0].y)
- number_of_cells_y as f64 * cell_side_length_padded);
use itertools::iproduct;
iproduct!(0..number_of_cells_x, 0..number_of_cells_y)
.map(|(i, j)| {
let corner = (
start_x + (i as f64) * cell_side_length_padded,
start_y + (j as f64) * cell_side_length_padded,
);
let points = nalgebra::SMatrix::<f64, 4, 2>::from_row_iterator([
corner.0,
corner.1,
corner.0 + cell_side_length,
corner.1,
corner.0 + cell_side_length,
corner.1 + cell_side_length,
corner.0,
corner.1 + cell_side_length,
]);
let cell_boundary_lengths = nalgebra::SVector::<f64, 4>::from_iterator(
(0..2 * 4).map(|_| cell_side_length),
);
VertexMechanics2D {
points,
velocity: nalgebra::SMatrix::<f64, 4, 2>::zeros(),
cell_boundary_lengths,
spring_tensions: nalgebra::SVector::<f64, 4>::from_element(spring_tensions),
cell_area,
central_pressure,
damping_constant,
diffusion_constant,
}
})
.collect::<Vec<_>>()
}
}
impl<const D: usize> VertexMechanics2D<D> {
/// Calculates the outer circle radius of the Regular Polygon given its area.
pub fn outer_radius_from_cell_area(cell_area: f64) -> f64 {
// let segment_length = Self::calculate_boundary_length(cell_area) / D as f64;
// segment_length / (std::f64::consts::PI / D as f64).tan() / 2.0
let boundary_length = Self::calculate_boundary_length(cell_area);
Self::outer_radius_from_boundary_length(boundary_length)
}
/// Calculates the outer circle radius of the Regular Polygon given its boundary length.
pub fn outer_radius_from_boundary_length(boundary_length: f64) -> f64 {
let segment_length = boundary_length / D as f64;
segment_length / (std::f64::consts::PI / D as f64).sin() / 2.0
}
/// Calculates the inner circle radius of the Regular Polygon given its area.
pub fn inner_radius_from_cell_area(cell_area: f64) -> f64 {
let boundary_length = Self::calculate_boundary_length(cell_area);
Self::inner_radius_from_boundary_length(boundary_length)
}
/// Calculates the inner circle radius of the Regular Polygon given its boundary length.
pub fn inner_radius_from_boundary_length(boundary_length: f64) -> f64 {
let segment_length = boundary_length / D as f64;
segment_length / (std::f64::consts::PI / D as f64).tan() / 2.0
}
}
impl<const D: usize>
Mechanics<
nalgebra::SMatrix<f64, D, 2>,
nalgebra::SMatrix<f64, D, 2>,
nalgebra::SMatrix<f64, D, 2>,
> for VertexMechanics2D<D>
{
fn calculate_increment(
&self,
force: nalgebra::SMatrix<f64, D, 2>,
) -> Result<(nalgebra::SMatrix<f64, D, 2>, nalgebra::SMatrix<f64, D, 2>), CalcError> {
// Calculate the total internal force
let middle = self.points.row_sum() / self.points.shape().0 as f64;
let current_ara: f64 = 0.5_f64
* self
.points
.row_iter()
.circular_tuple_windows()
.map(|(p1, p2)| p1.transpose().perp(&p2.transpose()))
.sum::<f64>();
let mut internal_force = self.points * 0.0;
for (index, (point_1, point_2, point_3)) in self
.points
.row_iter()
.circular_tuple_windows::<(_, _, _)>()
.enumerate()
{
let tension_12 = self.spring_tensions[index];
let tension_23 = self.spring_tensions[(index + 1) % self.spring_tensions.len()];
let mut force_2 = internal_force.row_mut((index + 1) % self.points.shape().0);
// Calculate forces arising from springs between points
let p_21 = point_2 - point_1;
let p_23 = point_2 - point_3;
let force1 =
p_21.normalize() * (self.cell_boundary_lengths[index] - p_21.norm()) * tension_12;
let force2 = p_23.normalize()
* (self.cell_boundary_lengths[(index + 1) % self.cell_boundary_lengths.len()]
- p_23.norm())
* tension_23;
// Calculate force arising from internal pressure
let middle_to_point_2 = point_2 - middle;
let force3 = middle_to_point_2.normalize()
* (self.cell_area - current_ara)
* self.central_pressure;
// Combine forces
force_2 += force1 + force2 + force3;
}
let dx = self.velocity;
let dv = force + internal_force - self.damping_constant * self.velocity;
Ok((dx, dv))
}
fn get_random_contribution(
&self,
rng: &mut rand_chacha::ChaCha8Rng,
dt: f64,
) -> Result<(nalgebra::SMatrix<f64, D, 2>, nalgebra::SMatrix<f64, D, 2>), RngError> {
let mut dvel = nalgebra::SMatrix::<f64, D, 2>::zeros();
let dpos = nalgebra::SMatrix::<f64, D, 2>::zeros();
if dt != 0.0 {
let random_vector: SVector<f64, 2> = wiener_process(rng, dt)?;
dvel.row_iter_mut().for_each(|mut r| {
r *= 0.0;
r += random_vector.transpose();
});
Ok((dpos, self.diffusion_constant * dvel))
} else {
Ok((nalgebra::SMatrix::<f64, D, 2>::zeros(), dvel))
}
}
}
impl<const D: usize> cellular_raza_concepts::Position<nalgebra::SMatrix<f64, D, 2>>
for VertexMechanics2D<D>
{
fn pos(&self) -> nalgebra::SMatrix<f64, D, 2> {
self.points
}
fn set_pos(&mut self, pos: &nalgebra::SMatrix<f64, D, 2>) {
self.points = *pos;
}
}
impl<const D: usize> cellular_raza_concepts::Velocity<nalgebra::SMatrix<f64, D, 2>>
for VertexMechanics2D<D>
{
fn velocity(&self) -> nalgebra::SMatrix<f64, D, 2> {
self.velocity
}
fn set_velocity(&mut self, velocity: &nalgebra::SMatrix<f64, D, 2>) {
self.velocity = *velocity;
}
}
#[cfg(test)]
mod test_vertex_mechanics_6n {
#[test]
fn test_fill_too_small() {
use crate::VertexMechanics2D;
use nalgebra::Vector2;
let models = VertexMechanics2D::<6>::fill_rectangle_flat_top(
200.0,
0.0,
0.0,
0.0,
0.0,
[Vector2::from([1.0, 1.0]), Vector2::from([2.0, 2.0])],
);
assert_eq!(models.len(), 0);
}
#[test]
fn test_fill_multiple() {
use crate::VertexMechanics2D;
use cellular_raza_concepts::Position;
use nalgebra::Vector2;
let models = VertexMechanics2D::<6>::fill_rectangle_flat_top(
36.0,
0.0,
0.0,
0.0,
0.0,
[Vector2::from([0.0; 2]), Vector2::from([100.0; 2])],
);
use itertools::Itertools;
for (m1, m2) in models.into_iter().circular_tuple_windows() {
if m1.pos().row_mean().transpose().x == m2.pos().row_mean().transpose().x {
let max = m1
.pos()
.row_iter()
.map(|row| row.transpose().y)
.max_by(|x0, x1| x0.partial_cmp(x1).unwrap())
.unwrap();
let min = m2
.pos()
.row_iter()
.map(|row| row.transpose().y)
.min_by(|x0, x1| x0.partial_cmp(x1).unwrap())
.unwrap();
assert!((max - min).abs() < 1e-7);
}
}
}
}