Initial data for general relativistic SPH with Centroidal Voronoi Tessellations

Abstract

In this work we present an alternative method to obtain a distribution of particles over an hyper surface, such that they obey a rest-mass density distribution $\rho (x^i)$. We use density profiles that can be written as $\rho (x^1,x^2,x^3)=\rho (x^1)\rho (x^2)\rho (x^3)$ in order to be able to use them as a probability density functions. We can find the relation between the chart $x^j$ and a uniform random variable ${\bar{x}}^j\in (0,1)$, say $F(x^j)={\bar{x}}^j$. Using the inverse of this function we relate a set of $N$ arbitrary number of points inside a cube with coordinates $\{x^j=F^{-1}({\bar{x}}^j)\}$ giving the position in order to get the density distribution $\rho (x^j)$. We get some noise due to the random distribution and we can notice that each time we relax the configuration on the cube we also get a better distribution of the desired physical configuration described with $\rho (x^j)$. This relaxation of the position of the particles in the cube has been performed a Lloyd's algorithm in 3D and we have used {\it Voro++} library in order to get the Voronoi tessellations.

References in corpus (1)

• Conservative, special-relativistic smoothed particle hydrodynamics