ericksenlesliex

FEniCSx implementation of numerical methods for the Ericksen--Leslie equations for nematic liquid crystal flow.

Governing equations

The governing equations in a general form can be stated as:

$$ \begin{aligned} \partialt v + ( v \cdot \nabla) v + \nabla p + \nabla \cdot T^E - \nabla \cdot T^L = 0, \ \nabla \cdot v = 0, \ d \times (\partialt d + ( v \cdot \nabla) d - {(\nabla v)} _{skw} d + \lambda {(\nabla v)} _{sym} d - \Delta d ) =0, \ \vert d \vert^2 = 1. \end{aligned} $$

Depending on the choice of complexity for the different stress tensors, we consider the following submodels.

Submodel 1

$$ \begin{aligned} \partialt v + ( v \cdot \nabla) v + \nabla p + \nabla \cdot [\nabla d]^T ( I - d \otimes d ) \Delta d - \nu \Delta v = 0, \ \nabla \cdot v = 0, \ \partialt d + ( I - d \otimes d ) [ ( v \cdot \nabla) d - \Delta d ] =0, \ \vert d \vert^2 = 1. \end{aligned} $$

Submodel 2

$$ \begin{aligned} \partialt v + ( v \cdot \nabla) v + \nabla p + \nabla \cdot [\nabla d]^T ( I - d \otimes d ) \Delta d - \nabla \cdot T^L = 0, \ \nabla \cdot v = 0, \ \partialt d + ( I - d \otimes d ) [ ( v \cdot \nabla) d - {(\nabla v)} _{skw} d + ( \lambda {(\nabla v)} _{sym} d - \Delta d ) ] =0, \ \vert d \vert^2 = 1. \end{aligned} $$

Numerical Methods

Currently the following numerical schemes are available. - linear Continuous Galerkin (CG) scheme fulfilling a discrete energy law - with mass-lumping - with a projection step (LhP) - without projection step (Lh) - with the standard $L^2$ inner product - with a projection step (LL2P) - without projection step (LL2) - linear Discontinuous Galerkin (DG) scheme fulfilling a discrete energy law - with a projection step (lpdg) - without projection step (ldg) - a linearized, decoupled fixed point solver to approximate a fully implicit Continuous Galerkin (CG) scheme (FPhD) - with mass-lumping, --> fulfills the unit-norm constraint exactly at every node of the mesh

All of the above methods use P2-P1-Taylor-Hood Finite Elements for velocity and space and CG1 elements for the director field and its discrete Laplacian.

Getting Started and Usage

All arguments to run simulations are given via the command line input. To see the options run the following command in the package directory:

python -m sim -h

Another usage example with several arguments:

python -m sim -m "LhP" -e spiral -cp -vtx -dh 20 -dt 0.01 -tur 0 -fsr 0.05 -sid "spiral-experiment" -T 0.05

Presets for several simulations --- usually in the context of publications --- are given in the folder 'sim/sim_presets/' usually in the form of a bash or python file. Examples for usage:

python -m sim.sim_presets.projection-spiral and sim/sim_presets/spiral.sh

Requirements

All requirements can be found in the file requirements.txt and can be installed via pip by

pip install -r requirements.txt

or via conda by

conda create --name my-env-name --file requirements.txt -c conda-forge

Notes

• Experiments of the class 'spiral' need a predefined mesh in .msh or .xdmf format in the folder 'input/meshes'.
• Results are automatically named and saved in the folder 'output/'

References to relevant publications

List references with links to publications this code was used for: - The decoupled fixed point solver is based on [1].

[1] Lasarzik, R., Reiter, M.E.V. Analysis and Numerical Approximation of Energy-Variational Solutions to the Ericksen–Leslie Equations. Acta Appl Math 184, 11 (2023). https://doi.org/10.1007/s10440-023-00563-9

Authors

• Maximilian E. V. Reiter, https://orcid.org/0000-0001-9137-7978

License

This project is licensed under the MIT License - see the LICENSE file for details