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cahn-hilliard-tumour-fenics

A Finite Element Approximation of a Cahn--Hilliard Tumour Model with FEniCS, by Dennis Trautwein (2020).

DOI of the original tumour model of Garcke, Lam, Sitka and Styles:

https://doi.org/10.1142/S0218202516500263

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General information about FEniCS:

a) Installation of FEniCS: See e.g. the following link:

https://fenicsproject.org/download/

b) Running FEniCS: Run a FEniCS program with the command:

python3 [name].py

c) Visualization: VTK-/VTU-/PVD- output files can be visualized with the program PARAVIEW. See e.g. the following link for download and installation:

https://www.paraview.org/download/

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Directories:

  1. The program 'ch_tumour_simple.py' is a simplified version of our programs. It is a modification of the demo 'demo_cahn-hilliard.py' from the FEniCS homepage:

https://fenicsproject.org/docs/dolfin/latest/python/demos.html

  1. The directory 'program_1D_convergence' contains the programs solving the Cahn--Hilliard tumour system and computing convergence rates in one dimension on the interval (0,1).

  2. The directory 'program_2D' contains the programs solving the Cahn--Hilliard tumour system in two dimensions on the square (0,12.5)^2.

  3. The directory 'program_3D' contains the programs solving the Cahn--Hilliard tumour system in three dimensions on the cube (0,3)^3.

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Programs:

i) In each directory you can run the program with the command:

python3 main_program.py

ii) The file 'settings.py' contains all settings: Parameters for the finite element discretization (mesh size, polynom degree, quadrature degree etc.) and other settings, e.g. for saving the output (formats like .vtu, .xml, .hdf4), handling initial values from a file, etc.

iii) The file 'class_Tumour.py' contains all model parameters and initializes the weak formulation of the system of partial differential equations.

iv) The file 'useful_functions.py' contains all functions needed for the program.

v) The file 'main_convergence_rate.py' (see the directory 'program_1D_convergence') can calculate the errors in some specific norms and the experimental orders of convergence (EOC), and save them into files. You can run the program with the command:

python3 main_convergence_rate.py

Note: Before running 'main_convergence_rate.py', you have to generate the data of solutions with different mesh sizes 'h' and you have to choose a reference solution. You can change the mesh size, polynom degree, quadrature degree in 'settings.py'.