T3: Density Estimates:
particle distributions and kernels

Before the tutorials:

As discussed in the Lecture, kernel density estimates are the key pillar of smoothed particle hydrodynamics (for an general overview on density estimations, see this survey of existing methods. Kernel density estimators are used to smooth a discretized distribution of tracer particles to obtain continuous fields needed in hydrodynamics. Therefore it is important to understand the intrinsic properties of such kernel density estimators.
Start with this picture

taken from this paper, figure 1 and think about:

• How are the simple particle distributions (especially Cubic Lattice and Hexagonally Close Packing) defined?
• What is a Glass distribution?
• How can you extend the program from T00 you used to set up a Cubic Lattice to the Hexagonally Close Packing) case?

The global properties of kernels have a significant impact on the performance of SPH. Make yourself familiar with some common choices of kernels (especially the spline and Wendland ones), as summarized in table 1 and figure 1 of this article.

During the tutorials:

You can start from the configuration of the code as obtained in T00 and collect some different particle distributions.

• Copy over the code and the parameter file as described in T00
• Copy over the Cubic Lattice you created in T00
• Copy over the Glass file provided cp $HOME/Hydro/glass_10x10x10 .

You can now run the program using different kernels and different neighbour numbers. To do so, you can:

• If no kernel is specified in the Config.sh file, the standard, Cubic Spline kernel is used.
• Alternatively, you can set QUINTIC_KERNEL, WENDLAND_C2_KERNEL, WENDLAND_C4_KERNEL, or WENDLAND_C6_KERNEL in the Config.sh file to to different kernels. Do not forget to re-compile after changing the kernel.
• To change the number of neighbours used, change the value for the DesNumNgb parameter in the box.param file, and re-run the simulation.

Now you can compare the resulting densities for the different particle distribution.

• Plot the distribution of densities for all particles in one simulation
• Repeat this for the different choices of kernels and number of neighbours
• Compute the mean value (and compare it to the one chosen for creating the particle distribution
• Compute the spread of densities for the different particles for each simulations
• Try to produce a similar figure than figure 3 of the above article.

Programming goals for T3:

Goal of this tutorial is that you learn better how to change the setup of the simulation and parameters.
What do you have to change to get only one output for each simulations?
Can you create a script to run the test with varying parameters?
How to produce a plot with multiple lines from different simulations?
How to produce one graph from the result of different simulations?

Solutions

• Example(s) for initial conditions
  • cp $HOME/Hydro/grid_10x10x10 .
    Make sure you are using BoxSize 1 in the parameter file!
  • cp $HOME/Hydro/packed_12x12x12 .
    Make sure you are using BoxSize 1 in the parameter file!
  • cp $HOME/Hydro/glass_10x10x10 .
    Make sure you are using BoxSize 100000 in the parameter file!
    Make sure you are using BoxSize 1 in the parameter file!^{[updated Nov 18]}
• Examples for script to run simulations
  • parameterfile with placeholders
  • shell script to run several simulations
• Example(s) for visualising the results
  • Use the example in the IDL language
    • idl -e show_kernels
  • Use the example in the Fortran language and gnuplot
    • ifort -g -traceback -check all -fpe0 -o meandensity meandensity.f90
    • bash runsims^{[updated Nov 15]}
    • gnuplot meandensity.plt
    • gv meandensity.ps
    • gnuplot density.plt
• You should see an image like this, with measured bias of the density estimate:
  
• Examples for script to set up a packed lattice
  • close_packed_lattice_new.pro in IDL