SPH Classes¶

Overview¶

The SPH classes implement Smooth Particle Hydrodynamics (SPH) density estimation and field interpolation using KDTree-based nearest-neighbour searches.

Supported dimensions:

SPH1D — 1D particle systems
SPH2D — 2D particle systems
SPH3D — 3D particle systems

All classes support:

Adaptive smoothing lengths based on nearest neighbours
Density estimation
Field interpolation
Optional particle masses
Cubic spline kernel

The general SPH estimator follows:

\[ \begin{align}\begin{aligned}\rho(\mathbf{x}) = \sum_i m_i W(|\mathbf{x} - \mathbf{x}_i|, h)\\f(\mathbf{x}) = \frac{\sum_i m_i f_i W(|\mathbf{x} - \mathbf{x}_i|, h)} {\sum_i m_i W(|\mathbf{x} - \mathbf{x}_i|, h)}\end{aligned}\end{align} \]

where W is the cubic SPH kernel and h is the smoothing length.

SPH1D¶

The SPH1D class performs SPH estimation in one dimension.

Initialization¶

sph = fiesta.sph.SPH1D()

By default, the cubic kernel is used:

sph.kernel_type = "cubic"

Setup¶

sph.setup(k=20, mass=None)

Parameters:

kint
Number of nearest neighbours used for smoothing
massarray, optional
Particle masses (defaults to 1 if None)

Assigning Mass¶

sph.assign_mass(mass)

If no mass is provided:

sph.mass = np.ones(len(points))

Density Estimation¶

dens = sph.get_density(x)

Returns SPH density evaluated at positions x.

Field Interpolation¶

sph.set_field(f)
f_est = sph.estimate(x)

f must be defined at particle positions
Returns SPH-smoothed field values

Core Method¶

sph.sph_estimate(x, f=None, dens=None)

If f is None → returns density If f is provided → returns interpolated field

SPH2D¶

The SPH2D class performs SPH estimation in two dimensions.

Initialization¶

sph = fiesta.sph.SPH2D()

Setup¶

sph.setup(k=20, mass=None)

Arguments:

k : number of nearest neighbours
mass : optional particle masses

Kernel¶

The SPH kernel is evaluated in 2D:

sph.kernel(r, h)

Uses cubic spline kernel:

\[W(r, h) = W_{\mathrm{cubic}}^{2D}(r, h)\]

Density Estimation¶

dens = sph.get_density(x, y)

Field Interpolation¶

sph.set_field(f)
f_est = sph.estimate(x, y)

Core Method¶

sph.sph_estimate(x, y, f=None, dens=None)

Returns:

density if f is None
interpolated field otherwise

SPH3D¶

The SPH3D class performs SPH estimation in three dimensions.

Initialization¶

sph = fiesta.sph.SPH3D()

Setup¶

sph.setup(k=50, mass=None)

Parameters:

k : number of nearest neighbours (default: 50)
mass : optional particle mass array

Kernel¶

sph.kernel(r, h)

Uses cubic kernel in 3D:

\[W(r, h) = W_{\mathrm{cubic}}^{3D}(r, h)\]

Density Estimation¶

dens = sph.get_density(x, y, z)

Field Interpolation¶

sph.set_field(f)
f_est = sph.estimate(x, y, z)

Core Method¶

sph.sph_estimate(x, y, z, f=None, dens=None)

Behavior:

If f is None → returns density
If f is provided → returns interpolated field

Algorithm Details¶

Nearest Neighbour Search¶

All SPH classes rely on KDTree queries:

nind, ndist = sph.nearest(..., k=self.k, return_dist=True)

nind: indices of nearest neighbours
ndist: distances to neighbours

Smoothing Length¶

For each query point:

h = max(ndist)

This defines the adaptive smoothing radius.

Kernel Weighting¶

Each contribution is weighted:

w_i = m_i * W(r_i, h)

Density or field is computed via summation.

Performance Notes¶

Complexity: O(N log N) for KDTree queries
Best performance with vectorized mass arrays

Usage Example¶

sph = fiesta.sph.SPH2D()
sph.setup(k=30)

sph.points = np.column_stack([x, y])
sph.assign_mass()

dens = sph.get_density(x, y)

sph.set_field(f)
f_interp = sph.estimate(x, y)