MLS Weight Functions

A weight function sets how strongly a neighbour influences the MLS fit at a particle, and its cutoff radius defines the neighbourhood. Weights are evaluated once per neighbour pair during the neighbour search (and again by refresh_geometry! on moving grids) and stored in pg.neighbor.weights.

Built-in weights

ExponentialWeightFunction(alpha, range) approximates

\[w(r) \approx \exp\!\left(-\alpha\, \frac{r^2}{R^2}\right), \qquad R = \texttt{range}.\]

The inverse range squared is precomputed. The exponential is evaluated as the reciprocal of a fourth-order Taylor polynomial of $e^{y}$, $y = \alpha r^2/R^2 \ge 0$. This is positive, monotonically decreasing and cheap, but it decays more slowly than the true exponential for large $\alpha r^2/R^2$.

InverseWeightFunction(alpha, range) evaluates $w(r) = 1/(r^2 + 10^{-12})$. The offset keeps coincident particles finite. alpha is stored but not used by the current implementation.

For both, range is the absolute cutoff radius $R$. A common choice is $R \approx 2.5$–$3.5\,\Delta x$. Higher-order schemes need more neighbours: a degree-$p$ fit in $D$ dimensions needs at least $\binom{p+D}{D} - 1$ neighbours, and schemes drop to a lower degree when a neighbourhood is too small.

Writing your own

Subtype MLSWeightFunction and provide two methods:

struct MyWeight{T} <: MLSWeightFunction
    range::T
end

# Called with the SQUARED distance
@inline (w::MyWeight{T})(dist_sq::Real) where {T} = ...

# Cutoff radius used for the neighbour search and bins
@inline HyperCloudCore.get_cutoff(w::MyWeight) = w.range

The weight must be positive within the cutoff. It does not need to vanish at the cutoff, since pairs beyond R are never listed. On moving grids, refresh_geometry! sets the weight of pairs that drift beyond R to exactly zero.

Reference

HyperCloudCore.ExponentialWeightFunction — Type

ExponentialWeightFunction{T} <: MLSWeightFunction
ExponentialWeightFunction(alpha::T, range::T)
(w::ExponentialWeightFunction)(dist_sq::Real) -> T

An exponential weighting function for Moving Least Squares (MLS) algorithms. It pre-computes the inverse square of the spatial range upon instantiation to optimize repeated distance evaluations.

Constructors

ExponentialWeightFunction(alpha::T, range::T)

alpha: Scaling parameter controlling the decay rate.
range: The spatial cutoff radius.

Callable / Functor

Evaluates the weight for a given squared distance dist_sq. It dispatches through a highly accurate, stable 4th-order polynomial approximation of exp(x) tailored strictly for negative inputs (x <= 0), shifting the expansion entirely into the denominator to eliminate standard library overhead.

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HyperCloudCore.ExponentialWeightFunction — Method

(w::ExponentialWeightFunction)(dist_sq::Real)

Evaluates the exponential weight using the pre-computed inverse range squared and dispatches it through the fast_exp_accurate numerical approximation.

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HyperCloudCore.InverseWeightFunction — Type

InverseWeightFunction{T} <: MLSWeightFunction
InverseWeightFunction(alpha::T, range::T)
(w::InverseWeightFunction)(dist_sq::Real) -> T

An inverse distance weighting function for Moving Least Squares (MLS) algorithms.

Constructors

InverseWeightFunction(alpha::T, range::T)

alpha: Scaling parameter.
range: The spatial cutoff radius.

Callable / Functor

Evaluates the weight for a given squared distance dist_sq. It uses a fixed 1e-12 tolerance offset in the denominator to prevent singularities and guarantee numerical stability upon coincident particle distances.

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HyperCloudCore.InverseWeightFunction — Method

(w::InverseWeightFunction)(dist_sq::Real)

Evaluates the inverse distance weight natively, utilizing a fixed 1e-12 tolerance offset to guarantee numerical stability upon coincident particle distances.

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