Tests and examples provided

All the scripts referred here can be found in the Examples folder of the Github repo, or also in local CALiPPSO folder (located in the Julia's package directory, e.g. .julia/packages in Linux). However, maybe it's better to just download the scripts into a location you find more convenient. The instructions below assume that you are executing julia inside the folder where the scripts are located.

In any case, two of the scripts are designed to show how CALiPPSO works in different dimensions and also the influence of the initial condition. A third one is designed to test the different solvers, but to run it you'll need to have all of them already installed. And a final one is just a simple script to exemplify how you can analyze the performance of CALiPPSO.jl.

Trying CALiPPSO in different dimensions

These tests are designed to jam monodisperse packings in $d=3, 4, 5$ dimensions to show that (i) CALiPPSO is a versatile algorithm; and, (ii) that produce_jammed_configuration! seamlessly works on different dimensions without tunning a bunch of parameters. Instead, the number of dimensions is simply inferred from the fixed size of the particles' position array, implemented as an SVector.

In both cases, the default solver used is GLPK. But we have also included (and commented out) the necessary lines to run the tests using Gurobi. This latter solver is much faster and more precise, as mentioned before.

Highly compressed initial configuration

First, the case where CALiPPSO works better is, naturally, when the initial condition is already close to its jamming point. (See the Theory behind CALiPPSO for more details.) Thus, for these examples, configurations of $N=1024$ particles have been compressed up to a pressure of $p=10^5$ using the Lubachevsky–Stillinger algorithm (see here for few more details).

To run this example, simply type the following line from a terminal

julia jamming-monodisperse-after-LS.jl

or, from inside the Julia's REPL, Jupyter Notebook, etc,


Then, produce_jammed_configuration! will be used to jam the configurations described before, and some output will be printed in the screen, showing the progress of CALiPPSO, etc. (Here you find more info about how to interpret such output.) For each $d$, once CALiPPSO converges, the program also prints the values of $sqrt{\Gamma^\star}-1$ and $max_{i,\mu} |\mathbf{s}_{i,\mu}^\star|$ obtained at each iteration, and some statistics of the times needed to solve the LOP instances. Furthermore network_of_contacts is also called on the jammed packing in every dimension. Thus, you can test that CALiPPSO works properly, not only reaching the jamming point, but also extracting the full network of contacts of the jammed configuration.


Note that to run this script you also need the files from where the initial condition is read. The relevant dat files are also contained in the Examples folder of the repo.

Low density initial configuration

The second example in which we test our algorithm is very similar to the previous, but now using an initial condition with low density. More precisely, this case jams configurations of $N=512$ particles, initialized with the particles placed at random, and with initial densities $\varphi_0 = \{0.3, 0.15, 0.1\}$, respectively for each dimension. Such initial conditions are produced using generate_random_configuration, which assigns uniformly random positions to the particles' centers(avoiding overlaps, of course).

To run this example, simply type the following line from a terminal

julia jamming-monodisperse-random-init_conf.jl 

or, from inside the Julia's REPL, Jupyter Notebook, etc,


The script works essentially as the previous one, also calling network_of_contacts once a packing has been jammed. Note however that it takes much longer to finish because much more LP optimizations are needed, as expected. Besides, the seed for the Julia's random number generator is initialized at the beginning of the script, for reproducibility. But it is not strictly necessary for the program to run.

Consider that running this script and similar programs might produce errors. Therefore, we do not guarantee that CALiPPSO can be used without errors with these types of initial conditions. It will always be better to initialize our algorithm with a density close to $\varphi_J$. In fact, note the following


Given that the initial condition is very far from the jamming point (i.e. $(\varphi_J - \varphi_0)/\varphi_J \sim \mathcal{O}(1)$), the first instances of the jamming LOP might lead to very large displacements, or to particles being block by $s_{bound}$ (i.e. the bound on $|\mathbf{s}_{i,\mu}|$). And even though overlaps are unlikely to occur (but are certainly possible in this scenario), the main issue we observed is that at some point the solvers fail to find any solution to the LOP. Moreover, some other types of issues might occur, specially in high dimensions.

Testing different solvers

To show that it is easy to use CALiPPSO with different solvers we provide the script testing-different-solvers.jl. Here we test the Gurobi, HiGHS, Clp, GLPK, and COSMO solvers on a same configuration, namely the $N=1024$, highly compressed configuration in $d=3$ (i.e. one of the ones used in a previous example). Reading through this script should provide a good idea on how to choose and tune different solvers using keywords, as described before. For instance each solver's attributes and arguments are explicitly defined; so this script can be used as a rough guideline on how to set the appropriate parameters for a given solver.

!!! Note Only the GLPK solver is installed alongside CALiPPSO.jl. Thus, before running this script, be sure to have installed all these solvers. As described in this section, all of them (except Gurobi) are straightforward to install. Only Gurobi requires having a license and a manual installation of the solver itself. But it's also rather easy.

Once you have all the solvers installed, you can run the script from a terminal in the following way

julia testing-different-solvers.jl

or, from inside the Julia's REPL, Jupyter Notebook, etc,


The main function, produce_jammed_configuration!, is first precompiled with each solver. Then, it is called with the initial $d=3$ condition mentioned above, once for each solver, in the order: Gurobi, HiGHS, Clp, GLPK, and COSMO. For each of them, the program outputs some info of the CALiPPSO progress, since verbose=true.

When a packing has been created with all the solvers, some information about how different they are is provided. More precisely, they are compared by computing the difference in radii and the mean squared distance between the particles' positions, using the packing jammed with Gurobi as reference.

Besides, once CALiPPSO converges with a given solver, the times required to solve each LOP instance with said solver are also printed out. Thus, it's easy to compare their performance.


The pure-Julia optimizer, Hypatia.jl can also be added to the list of solvers to test (naturally, after installing it). However, we did not include it by default because in our tests we observed that it consumed about 8GB of memory, compared to less than 1GB using other solvers. Besides, it was a rather slow solver, so we did not included it to avoid waiting too long. However, as we already mentioned, it is likely that we did not choose correctly its parameters and better results can be obtained.


Using the COSMO.jl solver we observed that the LOP instances are solved with a much smaller accuracy, usually leading to non-isostatic packings. In fact, in our experience produce_jammed_configuration! rarely converges when using COSMO as solver (at least not for this system size). But once again, this might be caused by our poor choice of parameters for this solver.

Simple example of Benchmark

In benchmark-different-ds.jl we provide a script to benchmark the performance of CALiPPSO using the same highly compressed initial conditions mentioned above. In fact, both scripts are essentially the same, except that in this one a benchmark is performed, using the @benchmarkable (from the Benchmarktools.jl package) macro and by calling produce_jammed_configuration! on these configurations 20 different times (for each value of $d$), and setting verbose=false to avoid unnecessary, repeated output in screen.

Dependencies for this test

To run this test you also need to install Benchmarktools.jl –in order to be able to call @benchmarkable– and HiGHS.jl. We chose to use the latter solver so the 20 repetitions can be done much more rapidly than only using GLPK.

Naturally, if you don't want to install HiGHS and prefer the default behaviour of produce_jammed_configuration!, you can do so by calling this main function without specifying the solver, solver_attributes, and solver_args keywords.

Once you have all installed these dependencies, you can run the script from a terminal in the following way

julia benchmark-different-ds.jl

or, from inside the Julia's REPL, Jupyter Notebook, etc,