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......@@ -4,10 +4,29 @@
*.ipynb_checkpoints
# ignore python cache
*__pycache__*
*.__pycache__
*test/__pycache__
*fxdgm/__pycache__
*examples/__pycache__
*examples/ReproduableCode/__pycache__
*.pytest_cache*
# ignore Sphinx documentation output
*docs/build
# Ignore coverage files
.coverage
htmlcov
# Ignore pip folder
*egg-info/*
# Ignore venv files
venv
\ No newline at end of file
venv
# Ignore .eggs
*.eggs
*eggs*
*.egg-info
# Ignore build files
build
# Ignore lyx temp files
*.lyx~
*.lyx#
# Ignore .DS_Store
.DS_Store
\ No newline at end of file
......@@ -3,7 +3,7 @@ image: docker:20.10.16 # Define the Docker image
stages: # Define stages in the pipeline
- prepare
- build
# - test
- test
- deploy
prepare:
......@@ -38,24 +38,31 @@ build-docs:
- docs/build # Save the build output for later stages
expire_in: 12 month # Optional: Set how long to keep the artifacts (default: 30 days)
# # Job to test the implementation
# test:
# stage: test
# dependencies:
# - prepare
# image:
# name: $CI_REGISTRY_IMAGE:latest
# entrypoint: [""]
# tags:
# - docker
# script:
# - python -m pytest # Run the tests
# Job to test the implementation
test:
stage: test
dependencies:
- prepare
image:
name: $CI_REGISTRY_IMAGE:latest
entrypoint: [""]
tags:
- docker
script:
- pytest --cov=fxdgm --cov-report=term --cov-report=html tests/ # Run the tests and store coverage
artifacts:
paths:
- htmlcov # Save the coverage report
expire_in: 12 month # Optional: Set how long to keep the artifacts (default: 30 days)
coverage: '/^TOTAL.*\s+(\d+\%)$/'
# Job to deploy documentation to GitLab Pages
pages:
stage: deploy
dependencies:
- prepare
- build-docs
- test
image:
name: $CI_REGISTRY_IMAGE:latest
entrypoint: [""]
......@@ -63,6 +70,7 @@ pages:
- docker
script:
- mv docs/build public # Move the build output to the "public" directory
- mv htmlcov public # Move the coverage report to the "public" directory
artifacts:
paths:
- public # Files in the "public" folder will be deployed to GitLab Pages
......
# Start with a base image that includes conda
FROM continuumio/miniconda3
FROM continuumio/miniconda3:latest
# Set environment variables
ENV CONDA_DEFAULT_ENV=fenicsx-env
ENV CONDA_DEFAULT_ENV=base
ENV PATH /opt/conda/envs/${CONDA_DEFAULT_ENV}/bin:$PATH
ENV HOME /root
# Create the environment and install packages
RUN conda create --name ${CONDA_DEFAULT_ENV} python=3.12.3 -y && \
conda install -n ${CONDA_DEFAULT_ENV} -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 matplotlib=3.8.4 numpy=1.26.4 scipy=1.14.0 gcc=12.4.0 sphinx=7.3.7 myst-parser=4.0.0 sphinx-copybutton=0.5.2 sphinx-rtd-theme=3.0.1 pytest==8.3.3 -y
# Activate environment
SHELL ["conda", "run", "-n", "fenicsx-env", "/bin/bash", "-c"]
# FEniCSx backend + documentation and testing dependencies
# sqlite for coverage, it necessary
RUN conda install -n ${CONDA_DEFAULT_ENV} -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 gcc=12.4.0 sphinx=7.3.7 myst-parser=4.0.0 sphinx-copybutton=0.5.2 sphinx-rtd-theme=3.0.1 pytest=8.3.3 pytest-cov=6.0.0 sqlite=3.41.2 -y
WORKDIR /root
# Set the default environment on container start
# ENTRYPOINT ["conda", "run", "-n", "fenicsx-env", "/bin/bash", "-c"]
ADD . /fxdgm
RUN pip install --editable /fxdgm/.
CMD ["bash"]
# Reproducibility Repository for Numerical Treatment of a Thermodynamically Consistent Electrolyte Model (B.Sc. Thesis - Jan Habscheid)
# fxdgm
[![Pipeline Status](https://git.rwth-aachen.de/Jan.Habscheid/bsc-electrolytemodels/badges/main/pipeline.svg)](https://git.rwth-aachen.de/Jan.Habscheid/bsc-electrolytemodels/pipelines)
[![Documentation](https://img.shields.io/badge/docs-latest-blue)](https://janhab.pages.rwth-aachen.de/bsc-electrolytemodels/)
[![Pipeline Status](https://git.rwth-aachen.de/janhab/fxdgm/badges/main/pipeline.svg)](https://git.rwth-aachen.de/janhab/fxdgm/pipelines)
[![Documentation](https://img.shields.io/badge/docs-latest-blue)](https://janhab.pages.rwth-aachen.de/fxdgm/)
[![coverage report](https://git.rwth-aachen.de/JanHab/fxdgm/badges/main/coverage.svg)](https://janhab.pages.rwth-aachen.de/fxdgm/htmlcov)
[![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.13645296.svg)](https://doi.org/10.5281/zenodo.13645296)
[![GitLab Version](https://img.shields.io/badge/version-1.0-blue.svg)](https://git.rwth-aachen.de/jan.habscheid/bsc-electrolytemodels/-/tags)
[![License](https://img.shields.io/badge/license-GPLv3-blue)](https://git.rwth-aachen.de/Jan.Habscheid/bsc-electrolytemodels/-/blob/main/LICENSE?ref_type=heads)
[![GitLab Version](https://img.shields.io/badge/version-1.0-blue.svg)](https://git.rwth-aachen.de/janhab/fxdgm/-/tags)
[![License](https://img.shields.io/badge/license-GPLv3-blue)](https://git.rwth-aachen.de/janhab/fxdgm/-/blob/main/LICENSE?ref_type=heads)
## Thesis
*A nonlinear, mixed finite element solver for the DGM electrolyte model*
This repository contains the code to reproduce the results presented in the bachelor thesis: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model
Find the thesis at [https://doi.org/10.18154/RWTH-2024-09837](https://doi.org/10.18154/RWTH-2024-09837)
<img src="media/logo.svg" alt="Logo" width="200" />
### Abstract
## Physical Background
Batteries play a crucial role in the energy transition. The production of green energy depends on external factors. Storing energy in batteries is necessary to access green energy at any time.
The system, which is solved, refers to the original work, [Overcoming the shortcomings of the Nernst–Planck model](https://doi.org/10.1039/C3CP44390F), from Wolfgang Dreyer, Clemens Guhlke and Rüdiger Müller in 2013.\
This paper introduces a new, generalized Nernst-Planck model, which is thermodynamically consistent, as the classical Nernst-Planck model fails to predict the correct ion-concentrations close to the boundaries.
The open-source package [FEniCSx](https://fenicsproject.org/) was used for the numerical implementation.
Better optimized batteries are essential for the future. Lifetime, loading time, and energy loss are just some aspects that must be improved to prepare for a greener future. Numerical simulations are crucial to understanding and optimizing batteries' behavior. Those simulations enable researchers to test many different materials without considerable additional expenses to, for example, find the best combination of anions and cations.
## Main Features
The classical Nernst-Planck model for the ion transport in an electrolyte fails to predict the correct concentration in the boundaries of the electrolyte. This work will present and analyze a thermodynamically consistent electrolyte model with dimensionless units under isothermal conditions. A simplified version of the system for the one-dimensional equilibrium of an ideal mixture and the incompressible limit will be considered. The numerical implementation of the model with the open-source software FEniCSx will be discussed. Furthermore, the influence of different boundary conditions, material parameters, solvation, and compressibility on the electric potential, pressure, and ion concentration will be investigated, and the model will be compared with the classical Nernst-Planck model. Examples of the double layer capacity and electrolytic diode will be considered.
- Solving steady [DGM](https://doi.org/10.1039/C3CP44390F) model in dimensionless units
- for a ternary electrolyte (cations, anions, neutral solvent)
- for an electrolyte of N arbitrary species
- Local mesh refinement for one-dimensional domains towards the electrode
- [Testcases](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/tests?ref_type=heads) for the one-dimensional case or the two-dimensional electrolytic diode
- Solutions for the [Double-Layer Capacity](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode/DoubleLayerCapacity?ref_type=heads), both numerical and analytical
- [Numerical Convergence](https://git.rwth-aachen.de/JanHab/fxdgm/-/blob/main/examples/ReproducableCode/Convergence.py?ref_type=heads) with relaxation parameter for newtons method
- Two-dimensional testcases for the example of the [electrolytic diode](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode/ElectrolyticDiode?ref_type=heads)
## Installation
As a numerical solver, mainly FEniCSx was used and installed via conda.
All the calculations were performed on a Linux machine. According to the documentation, everything should work well on macOS, but this was not tested. FEniCSx offers some beta versions for Windows support, but it is recommended to use WSL2 instead.
Install the fxdgm package with pip to get all the implemented functions.
``` bash
pip install git+https://git.rwth-aachen.de/JanHab/fxdgm
```
conda create --name fenicsx-env python=3.12.3 -y
conda activate fenicsx-env
conda install -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 matplotlib=3.8.4 numpy=1.26.4 scipy=1.14.0 pytest==8.3.3 -y
```
### Alternative installation
Use the "environment.yml" file to install all necessary environments
For the backend, FEniCSx was used and installed via conda.
The necessery dependencies can be installed with
```
conda env create -f environment.yml
``` bash
conda install -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 gcc=12.4.0 -y
```
It is also possible to install the FEniCSx backend in a different manner. See the [FEniCSx documentation](https://fenicsproject.org/download/) for this.
Although this installation method should work, it was not tested for the purpose of this package.
### macOS installation using Docker
```
The docker installation method works for linux too. It was not tested on windows.
``` bash
docker compose build
docker compose run solver
```
### Testing
Use pytest with
```
For testing clone the repository, install pytest and run the tests with
``` bash
pip install pytest==8.3.3
python -m pytest
```
to verify that everything was installed correctly.
## Usage
Find the visualizations from the thesis and some extra calculations in the "examples" folder.
In the subfolder "ReproducableCode" is the code, to execute the calculations with some first visualizations.
The subfolder "Data" stores the data for all the simulations in a *.npz file, which can be read with numpy `np.load(file.npz)`.
"Visualizations" creates the necessary figures from the thesis and stores them in *.svg format in "Figures".
In "src" there are the generic FEniCSx implementations, that were used to calculate the examples.
Find the package source code in [fxdgm](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/fxdgm?ref_type=heads).
This implements the nonlinear electrolyte model.
Furthermore, some physical examples are provided in the [examples](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples?ref_type=heads).
In the subfolder [ReproducableCode](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode?ref_type=heads) is the code, to execute the calculations with some first visualizations.
The subfolder [Data](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/Data?ref_type=heads) stores the data for all the simulations in a *.npz file, which can be read with numpy `np.load(file.npz)`.
[Visualizations](https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/Visualizations?ref_type=heads) creates the necessary figures from the thesis and stores them either in *.svg or *.pdf format in "Figures".
## Contact
**Author**
### Author
- Jan Habscheid
- Jan.Habscheid@rwth-aachen.de
- [Jan.Habscheid@rwth-aachen.de](mailto:Jan.Habscheid@rwth-aachen.de)
**Supervisor**
- Dr. Lambert Theissen
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- theisen@acom.rwth-aachen.de
### Supervisor
**Supervisor**
- Dr. Lambert Theisen
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- [theisen@acom.rwth-aachen.de](mailto:theisen@acom.rwth-aachen.de)
- Prof. Dr. Manuel Torrilhon
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- mt@acom.rwth-aachen.de
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- [mt@acom.rwth-aachen.de](mailto:mt@acom.rwth-aachen.de)
......@@ -6,16 +6,16 @@
# -- Project information -----------------------------------------------------
# https://www.sphinx-doc.org/en/master/usage/configuration.html#project-information
project = 'Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model'
project = 'fxdgm'
copyright = '2024, Jan Habscheid'
author = 'Jan Habscheid'
release = 'September, 2024'
# Add src folder to path
# Add fxdgm folder to path
import os
import sys
sys.path.insert(0, os.path.abspath(os.path.join("..", "../src")))
sys.path.insert(0, os.path.abspath('../src'))
sys.path.insert(0, os.path.abspath(os.path.join("..", "../fxdgm")))
sys.path.insert(0, os.path.abspath('../fxdgm'))
# -- General configuration ---------------------------------------------------
# https://www.sphinx-doc.org/en/master/usage/configuration.html#general-configuration
......
Example
=======
| In this example, we are going to solve the thermodynamically consistent electrolyte model for a incompressible, ternary electrolyte, as it is done in "examples/ReproducableCode/TernaryElectrolyte.py"
| In this example, we are going to solve the thermodynamically consistent electrolyte model for a incompressible, ternary electrolyte, as it is done in `https://git.rwth-aachen.de/JanHab/fxdgm/-/blob/main/examples/ReproducableCode/TernaryElectrolyte.py?ref_type=heads <https://git.rwth-aachen.de/JanHab/fxdgm/-/blob/main/examples/ReproducableCode/TernaryElectrolyte.py?ref_type=heads>`_.
| The code will work if executed in the examples/ReproducableCode folder, as the TernaryElectrolyte.py file is located in the same folder
Import FEniCSx implementation and necessary libraries
-----------------------------------------------------
Install the package and necessary libraries
-------------------------------------------
.. code-block:: python
import sys
import os
pip install git+https://git.rwth-aachen.de/JanHab/fxdgm
conda install -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 gcc=12.4.0 -y
src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../..', 'src')
sys.path.insert(0, src_path)
Import the necessary libraries
-------------------------------------------
from Eq04 import solve_System_4eq
.. code-block:: python
del sys.path[0]
from fxdgm import solve_System_4eq
import matplotlib.pyplot as plt
import numpy as np
......
.. Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation master file, created by
sphinx-quickstart on Tue Sep 3 10:16:31 2024.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
fxdgm
=====
.. image:: https://git.rwth-aachen.de/janhab/fxdgm/badges/main/pipeline.svg
:target: https://git.rwth-aachen.de/JanHab/fxdgm/pipelines
.. image:: https://img.shields.io/badge/docs-latest-blue
:target: https://janhab.pages.rwth-aachen.de/fxdgm/
.. image:: https://zenodo.org/badge/DOI/10.5281/zenodo.13645296.svg
:target: https://doi.org/10.5281/zenodo.13645296
.. image:: https://img.shields.io/badge/version-1.0-blue.svg
:target: https://git.rwth-aachen.de/janhab/fxdgm/-/tags
*A nonlinear, mixed finite element solver for the DGM electrolyte model*
.. image:: ../../media/logo.svg
:width: 300px
Physical Background
===================
| The system, which is solved, refers to the original work, `Overcoming the shortcomings of the Nernst–Planck model <https://doi.org/10.1039/C3CP44390F>`_, from Wolfgang Dreyer, Clemens Guhlke and Rüdiger Müller in 2013.
| This paper introduces a new, generalized Nernst-Planck model, which is thermodynamically consistent, as the classical Nernst-Planck model fails to predict the correct ion-concentrations close to the boundaries.
| The open-source package `FEniCSx <https://fenicsproject.org/>`_ was used for the numerical implementation.
Reproducibility Repository for Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation
=====================================================================================================================
Main Features
=============
.. admonition:: \ \
- Solving stationary `DGM <https://doi.org/10.1039/C3CP44390F>`_ model in dimensionless units
This python package has the purpose to make the implementation and the examples from the bachelor thesis "Numerical Treatment of a Thermodynamically Consistent Electrolyte Model" reproducible. The code is written in Python and uses the finite element library FEniCSx. The code is available at: https://doi.org/10.5281/zenodo.13645296
- for a ternary electrolyte (cations, anions, neutral solvent)
Thesis
======
| This repository contains the code to reproduce the results presented in the bachelor thesis: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model
| Find the thesis at: https://doi.org/10.18154/RWTH-2024-09837
- for an electrolyte of N arbitrary species
Abstract
========
- Local mesh refinement for one-dimensional domains towards the electrode
| Batteries play a crucial role in the energy transition. The production of green energy depends on external factors. Storing energy in batteries is necessary to access green energy at any time.
- `Testcases <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/tests?ref_type=heads>`_ for the one-dimensional case or the two-dimensional electrolytic diode
| Better optimized batteries are essential for the future. Lifetime, loading time, and energy loss are just some aspects that must be improved to prepare for a greener future. Numerical simulations are crucial to understanding and optimizing batteries' behavior. Those simulations enable researchers to test many different materials without considerable additional expenses to, for example, find the best combination of anions and cations.
- Solutions for the `Double-Layer Capacity <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode/DoubleLayerCapacity?ref_type=heads>`_, both numerical and analytical
- `Numerical Convergence <https://git.rwth-aachen.de/JanHab/fxdgm/-/blob/main/examples/ReproducableCode/Convergence.py?ref_type=heads>`_ with relaxation parameter for newtons method
- Two-dimensional testcases for the example of the `electrolytic diode <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode/ElectrolyticDiode?ref_type=heads>`_
| The classical Nernst-Planck model for the ion transport in an electrolyte fails to predict the correct concentration in the boundaries of the electrolyte. This work will present and analyze a thermodynamically consistent electrolyte model with dimensionless units under isothermal conditions. A simplified version of the system for the one-dimensional equilibrium of an ideal mixture and the incompressible limit will be considered. The numerical implementation of the model with the open-source software FEniCSx will be discussed. Furthermore, the influence of different boundary conditions, material parameters, solvation, and compressibility on the electric potential, pressure, and ion concentration will be investigated, and the model will be compared with the classical Nernst-Planck model. Examples of the double layer capacity and electrolytic diode will be considered.
Installation
============
| As a numerical solver, mainly FEniCSx was used and installed via conda.
| All the calculations were performed on a Linux machine. According to the documentation, everything should work well on macOS, but this was not tested. FEniCSx offers some beta versions for Windows support, but it is recommended to use WSL2 instead.
| Install the fxdgm package with pip to get all the implemented functions.
.. code-block::
conda create --name fenicsx-env python=3.12.3 -y
conda activate fenicsx-env
conda install -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 matplotlib=3.8.4 numpy=1.26.4 scipy=1.14.0 -y
pip install git+https://git.rwth-aachen.de/JanHab/fxdgm
Alternative installation
========================
| Use the "environment.yml" file to install all necessary environments
| For the backend, FEniCSx was used and installed via conda.
| The necessary dependencies can be installed with
.. code-block::
conda env create -f environment.yml
conda install -c conda-forge fenics-dolfinx=0.8.0 mpich=4.2.1 pyvista=0.43.10 gcc=12.4.0 -y
| It is also possible to install the FEniCSx backend in a different manner. See the `FEniCSx documentation <https://fenicsproject.org/download/>`_ for this.
| Although this installation method should work, it was not tested for the purpose of this package.
macOS installation using Docker
-------------------------------
| The docker installation method works for linux too. It was not tested on windows.
.. code-block::
docker compose build
docker compose run solver
Testing
=======
| For testing clone the repository, install pytest and run the tests with
.. code-block::
pip install pytest==8.3.3
python -m pytest
Usage
=====
| Find the visualizations from the thesis and some extra calculations in the "examples" folder.
| In the subfolder "ReproducableCode" is the code, to execute the calculations with some first visualizations.
| The subfolder "Data" stores the data for all the simulations in a ".npz" file, which can be read with numpy `np.load(file.npz)`. "Visualizations" creates the necessary figures from the thesis and stores them in ".svg" format in "Figures".
| In "src" there are the generic FEniCSx implementations, that were used to calculate the examples.
| Find the package source code in `fxdgm <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/fxdgm?ref_type=heads>`_
| This implements the nonlinear electrolyte model.
| Furthermore, some physical examples are provided in the `examples <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples?ref_type=heads>`_ folder.
| In the subfolder `ReproducableCode <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/ReproducableCode?ref_type=heads>`_ is the code, to execute the calculations with some first visualizations.
| The subfolder `Data <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/Data?ref_type=heads>`_ stores the data for all the simulations in a \*.npz file, which can be read with numpy `np.load(file.npz)`.
| `visualizations <https://git.rwth-aachen.de/JanHab/fxdgm/-/tree/main/examples/Visualizations?ref_type=heads>`_ creates the necessary figures from the thesis and stores them either in \*.svg or \*.pdf format in "Figures".
Contact
=======
......@@ -59,22 +101,24 @@ Contact
**Supervisor**
- Dr. Lambert Theissen
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- theisen@acom.rwth-aachen.de
- Dr. Lambert Theisen
**Supervisor**
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- theisen@acom.rwth-aachen.de
- Prof. Dr. Manuel Torrilhon
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- mt@acom.rwth-aachen.de
- ACoM - Applied and Computational Mathematics
- RWTH Aachen University
- mt@acom.rwth-aachen.de
.. toctree::
src
src/oneD
src/ElectrolyticDiode
src/Miscellaneous
example
:maxdepth: 2
:caption: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:
:caption: fxdgm
:hidden:
Electrolytic Diode
===================
.. warning::
If the Newton solver diverges for any of the solutions, you may try to reduce the relaxation parameter.
.. autofunction:: ElectrolyticDiode.ElectrolyticDiode
Miscellaneous
=============
.. autofunction:: RefinedMesh1D.create_refined_mesh
.. automodule:: Helper_DoubleLayerCapacity
:members:
:undoc-members:
:show-inheritance:
\ No newline at end of file
Source Code
===========
1D Electrolyte
==============
.. warning::
If the Newton solver diverges for any of the solutions, you may try to reduce the relaxation parameter.
.. autofunction:: Eq04.create_refined_mesh
.. autofunction:: EqN.solve_System_Neq
.. autofunction:: Eq04.solve_System_4eq
.. autofunction:: Eq02.solve_System_2eq
.. autofunction:: ElectrolyticDiode.ElectrolyticDiode
.. autofunction:: EqN.solve_System_Neq
.. automodule:: Helper_DoubleLayerCapacity
:members:
:undoc-members:
:show-inheritance:
\ No newline at end of file
.. autofunction:: Eq02.solve_System_2eq
\ No newline at end of file
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