Welcome to my GitHub repository! Here you can find the numerical optima that appears in the paper https://arxiv.org/abs/2303.12389.
In order to use the files of this repository, you first have to clone it :
https://github.com/EloiMartinet/Neumann_Sphere.git
In both level-set and density cases, a FreeFem++ script is provided to compute the eigenvalues of the optimums, which are stored in MEDIT format. To execute the scripts, first install FreeFem++ (https://freefem.org/). You can then go into the Density folder in your command line and execute
FreeFem++ Neumann_Density.edp Sphere/mu_1/11.6
Note that we don't precise any extension, since the script will automatically load the *.mesh and *.sol files. Similarly, you can run the command
FreeFem++ Neumann_LS.edp Sphere/mu_2/7.121706429550119
from the Levelset folder to get the corresponding eigenvalues.
To visualize the results, you will need to install MEDIT (https://hal.inria.fr/inria-00069921/document) and from the Density folder, execute
medit Torus/mu_1/11.6
and then simply type the "m" key to show the density. You can also uncomment the line
// medit("Optimal density.", Th, rho, wait=1);
in the file Neumann_Sphere_Density.edp but be aware that the display takes a lot of time in this case.
The code to generate the optimal density for a target mass on a sphere can be found in Density/Sphere/code. The main file is optimizer.py. To be able to run it, you will need to install python and GetFEM (https://getfem.org/) for the finite elements computations, as well as cyipopt (https://cyipopt.readthedocs.io/en/stable/) for the interior point optimization procedure. You can then execute the file by executing
python3 optimizer.py
This will output the result of each iteration in medit format in the folder results. You can then visualize an animation of the iterations by executing
medit results/rho -a 0 N
where N is the number of iterations. The parameters of the optimization can be changed directly inside the main() function of optimizer.py (maximum number of iterations, regularization parameters, meshes used, etc).
The optimizer.py file needs some pre-computed derivatives (in derivatives) which depend on the GetFEM meshes (in meshes). The derivatives are provided, but you can re-compute them by executing
python3 martices_derivatives.py input_mesh output_derivative_file
Similar simulations on the plane can be found at https://github.com/EloiMartinet/Maximization_Of_Neumann_Eigenvalues.
In addition to the previous tools, you will need to install the Advection tool (https://github.com/ISCDtoolbox/Advection) which allows you to advect the level-set function. For remeshing, please install mmgtools (https://www.mmgtools.org/). For distance function related operations, please install mshdist (https://github.com/ISCDtoolbox/Mshdist).
Please navigate to the folder Levelset/Sphere/code. There is two main files : ersatz.py and remeshing.py, each corresponding to a method implemented in the paper. Executing
python3 ersatz.py
will output the meshes and level sets obtained by the ersatz procedure to the folder results. To visualize them, please run
medit results/ls -a 0 150
You can press C,E in the medit window to display the set (in yellow).
Executing
python3 remeshing.py
will do the same but with the remeshing procedure. The initial mesh is given in init.mesh but you can specify another one obtained by the previous algorithm, for instance results/ls.137.mesh.
Enjoy!
You can find more wonderful simulations at https://eloimartinet.github.io/ !