T2: 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 Hexagonal Close Packing) defined?
• What is a Glass distribution?
• How can you extend the program from T0b you used to set up a Cubic Lattice to the Hexagonal 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 T01 and collect some different particle distributions.

• Copy over the code and the parameter file as used in T01
• Copy over the Cubic Lattice you created in T01
• For people who want to look at something special: Take the setup_grid.py from T01 and modify it so that you get random positions of particles within the box.

You can now run the program using different kernels and different neighbor numbers. Keep in mind that you do not need to run the simulation for a very long time, basically you just need one output file. So add the StopAfterNSteps parameter in the box.param file and set it to 5. Every setup/combination of kernel, number of neighbors, and initial particle distribution is an independent simulation you need to perform. To do so, you can:

• Specify the kernel to be used in the Config.sh file. Choices are STANDARD_KERNEL (i.e., the standard, Cubic Spline kernel), alternatively you can set QUINTIC_KERNEL, WENDLAND_C2_KERNEL, WENDLAND_C4_KERNEL, or WENDLAND_C6_KERNEL in the Config.sh file to select different kernels. Only one of the above settings can be set. Do not forget to re-compile after changing the kernel.
• To change the number of neighbors used, change the value for the DesNumNgb parameter in the box.param file, and re-run the simulation.
  Hint: When ever you perform a simuation, the code is producing a parameters-usedvalues file, which also contains the default vales of unspecified parameters. So here you can check what is the default value of DesNumNgb for the kernel you have chosen!
• To change the particle distribution to be used, change the name given for the InitCondFile parameter in the box.param file to the names of the different distributions and re-run the simulation.

Now you can compare the resulting densities for the different simulations you produced. You can read the density ("RHO ") for all particles within the different simulations and compute the mean and the standard deviation, compared to the true density used to create the initial conditions. You can do this step by step:

• Plot the distribution of densities for all particles in one simulation.
• Compute the mean, median and the width of the distribution. Note down the values, together with the configuration used.
• Repeat this for the different choices of kernels, number of neighbors and initial particle distributions.
• Do some simple plots comparing these values for your different choices.
• Try to produce a figure similar to 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
  • wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/grid_10x10x10
    Make sure you are using BoxSize 1 in the parameter file!
  • wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/packed_12x12x12
    Make sure you are using BoxSize 1 in the parameter file!
  • wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/glass_10x10x10
    Make sure you are using BoxSize 1 in the parameter file!
• Example for script to read density and compute mean and variance:
  • wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T02/print_values.py
  • python3 print_values.py
• Examples for script to run simulations
  • You can put placeholders in the param file which can be replaced in a script, so that you can perfoem several simulations in a row. So for example, replace the value of SnapshotFileBase by wc2_XXXX and the vale of DesNumNgb by XXXX. An exampec can be found here:
    wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T02/box.ini
  • Then, you can modify your bash script to create the according paramerter file on the fly. An example can be found here:
    wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T02/run_sims.sh
  • With theis setup, you can start a simulation in the normal way:
    ./run_sims.sh
  • Now you can change your analythis scripts to loop over the different files.
• Example(s) for visualizing the results
  • wget https://www.usm.uni-muenchen.de/~dolag/Lille2026/T02/show_kernels.py
  • python3 show_kernels.py
• You should see part of an image like this, with measured bias of the density estimate:
  
  Now extend the simulations you do and modify the routines to use different kernels, different neighbor values and different particle distributions

Useful commands

• More informations on the OpenGadget3 containers
• Start your docker session:docker run -it -v :/mnt --name opengadget3_container opengadget3:XXX
• Connect from a second shell to a running docker session:docker exec -it opengadget3_container bash
• Get the simulation code:cp -r ~/OpenGadget3 .
• Copy the config file: wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/Config.sh
• Copy the parameter file: wget https://www.usm.uni-muenchen.de/~dolag/Hydro/Hydro/box.param
• Setup the environment for compiling the code: source Build/Container_build.sh
• Compile the code: make -j
• Run a simulation: mpiexec -np 2 OpenGadget3/OpenGadget3 box.param