SPH Approximation of a Function

Let

\prod f (x)

$\prod f (x)$ the integral approximation of a scalar function

f

$f$ in space:(1)

\prod f (x) = \int_{Ω} f (y) W (x - y, h) d y

$\prod f (x) = \int_{Ω} f (y) W (x - y, h) d y$

with

h

$h$ the so-called smoothing length and

W

$W$ a kernel approximation such that:(2)

\forall x, \int_{Ω} W (x - y, h) d y = 1

$\forall x, \int_{Ω} W (x - y, h) d y = 1$

and in a suitable sense(3)

\forall x, \lim_{h \to 0} W (x - y, h) = δ (x - y)

$\forall x, \lim_{h \to 0} W (x - y, h) = δ (x - y)$

$δ$ $δ$ denotes the Dirac function.

Let a set of particles

i

$i$ =1, n at positions

x_{i}

$x_{i}$ (

i

$i$ =1,n) with mass

m_{i}

$m_{i}$ and density

ρ_{i}

$ρ_{i}$ . The smoothed approximation of the function

f

$f$ is (summation over neighboring particles and the particle

i

$i$ itself):(4)

\prod_{s} f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) W (x - y, h)

$\prod_{s} f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) W (x - y, h)$

The derivatives of the smoothed approximation are obtained by ordinary differentiation.(5)

\nabla f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) \nabla W (x - y, h)

$\nabla f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) \nabla W (x - y, h)$

The following kernel ¹ which is an approximation of Gaussian kernel by cubic splines was chosen (Figure 1):(6)

r \leq h \Rightarrow W (r, h) = \frac{3}{2 π h^{3}} [\frac{2}{3} - {(\frac{r}{h})}^{2} + \frac{1}{2} {(\frac{r}{h})}^{3}]

$r \leq h \Rightarrow W (r, h) = \frac{3}{2 π h^{3}} [\frac{2}{3} - {(\frac{r}{h})}^{2} + \frac{1}{2} {(\frac{r}{h})}^{3}]$

(7)

h \leq r \leq 2 h \Rightarrow W (r, h) = \frac{1}{4 π h^{3}} {(2 - \frac{r}{h})}^{3}

$h \leq r \leq 2 h \Rightarrow W (r, h) = \frac{1}{4 π h^{3}} {(2 - \frac{r}{h})}^{3}$

and (8)

2 h \leq r \Rightarrow W (r, h) = 0

$2 h \leq r \Rightarrow W (r, h) = 0$

This kernel has compact support, so that for each particle $i$ $i$ , only the closest particles contribute to approximations at $i$ $i$ (this feature is computationally efficient). The accuracy of approximating Equation 1 by Equation 4 depends on the order of the particles.

Monaghan J.J., “Smoothed Particle Hydrodynamics”, Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.