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Jacob Hummel edited 3-Visualization.tex
about 8 years ago
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The particles in an SPH simulation are best thought of as fluid elements sampling the continuum properties of the gas they represent \citep{Lucy1977,GingoldMonaghan1977,Monaghan1992,Springel2010}. They accomplish this by serving as Lagrangian tracers over which the continuum properties are interpolated using a smoothing kernel $W$. While it is possible to use alternative kernels, most modern SPH implementations (including \textsc{gadget}) utilize a cubic spline kernel \citep{Springel2014}:
\begin{equation}
W(r,h) W(r,h_{\rm sm}) =
\begin{cases}
1 - 6 \left(
\frac{r}{h} \frac{r}{h_{\rm sm}} \right)^2 + 6 \left(
\frac{r}{h} \frac{r}{h_{\rm sm}} \right)^3, & 0 \leq
\frac{r}{h} \frac{r}{h_{\rm sm}} \leq \frac{1}{2}\\
2 \left(1 -
\frac{r}{h}\right)^3, \frac{r}{h_{\rm sm}}\right)^3, & \frac{1}{2} <
\frac{r}{h} \frac{r}{h_{\rm sm}} \leq 1\\
0, & \frac{r}{h} > 1,\\
\end{cases}
\end{equation}
where $r$ is the radius and
$h$ $h_{\rm sm}$ is the characteristic width of the kernel, otherwise known as the smoothing length. The physical density at any point, $\rho(\bf{r})$, is then represented by the sum over all particles
\begin{equation}
\rho({\bf r}) \simeq \sum_j m_j W({\bf r} - {\bf r}_j,
h), h_{\rm sm}),
\end{equation}
where $m_j$ is the mass of particle $j$, located at ${\bf r}_j$.
As such, creating visualizations that faithfully reproduce the actual density distribution requires performing this sum over all particles of interest; this can be quite computationally intensive depending on the number of particles involved and the desired resolution.