From 01096de68fc79213703b62acb9da5b681d0c77cc Mon Sep 17 00:00:00 2001
From: JanHab <Jan.Habscheid@web.de>
Date: Tue, 3 Sep 2024 11:42:06 +0200
Subject: [PATCH] Added Sphinx docs

---
 README.md                                     |    6 +-
 docs/Makefile                                 |   20 +
 docs/build/doctrees/environment.pickle        |  Bin 0 -> 40332 bytes
 docs/build/doctrees/example.doctree           |  Bin 0 -> 9368 bytes
 docs/build/doctrees/index.doctree             |  Bin 0 -> 18036 bytes
 docs/build/doctrees/src.doctree               |  Bin 0 -> 263188 bytes
 docs/build/html/.buildinfo                    |    4 +
 .../html/_images/TernaryConcentrations.svg    |  952 ++++++
 docs/build/html/_images/TernaryPotential.svg  |  756 +++++
 docs/build/html/_images/TernaryPressure.svg   |  679 +++++
 docs/build/html/_sources/example.rst.txt      |  105 +
 docs/build/html/_sources/index.rst.txt        |   82 +
 docs/build/html/_sources/src.rst.txt          |   20 +
 .../_sphinx_javascript_frameworks_compat.js   |  123 +
 docs/build/html/_static/basic.css             |  925 ++++++
 docs/build/html/_static/check-solid.svg       |    4 +
 docs/build/html/_static/clipboard.min.js      |    7 +
 docs/build/html/_static/copy-button.svg       |    5 +
 docs/build/html/_static/copybutton.css        |   94 +
 docs/build/html/_static/copybutton.js         |  248 ++
 docs/build/html/_static/copybutton_funcs.js   |   73 +
 docs/build/html/_static/css/badge_only.css    |    1 +
 .../_static/css/fonts/Roboto-Slab-Bold.woff   |  Bin 0 -> 87624 bytes
 .../_static/css/fonts/Roboto-Slab-Bold.woff2  |  Bin 0 -> 67312 bytes
 .../css/fonts/Roboto-Slab-Regular.woff        |  Bin 0 -> 86288 bytes
 .../css/fonts/Roboto-Slab-Regular.woff2       |  Bin 0 -> 66444 bytes
 .../_static/css/fonts/fontawesome-webfont.eot |  Bin 0 -> 165742 bytes
 .../_static/css/fonts/fontawesome-webfont.svg | 2671 +++++++++++++++++
 .../_static/css/fonts/fontawesome-webfont.ttf |  Bin 0 -> 165548 bytes
 .../css/fonts/fontawesome-webfont.woff        |  Bin 0 -> 98024 bytes
 .../css/fonts/fontawesome-webfont.woff2       |  Bin 0 -> 77160 bytes
 .../_static/css/fonts/lato-bold-italic.woff   |  Bin 0 -> 323344 bytes
 .../_static/css/fonts/lato-bold-italic.woff2  |  Bin 0 -> 193308 bytes
 .../html/_static/css/fonts/lato-bold.woff     |  Bin 0 -> 309728 bytes
 .../html/_static/css/fonts/lato-bold.woff2    |  Bin 0 -> 184912 bytes
 .../_static/css/fonts/lato-normal-italic.woff |  Bin 0 -> 328412 bytes
 .../css/fonts/lato-normal-italic.woff2        |  Bin 0 -> 195704 bytes
 .../html/_static/css/fonts/lato-normal.woff   |  Bin 0 -> 309192 bytes
 .../html/_static/css/fonts/lato-normal.woff2  |  Bin 0 -> 182708 bytes
 docs/build/html/_static/css/theme.css         |    4 +
 docs/build/html/_static/doctools.js           |  156 +
 .../html/_static/documentation_options.js     |   13 +
 docs/build/html/_static/file.png              |  Bin 0 -> 286 bytes
 docs/build/html/_static/jquery.js             |    2 +
 docs/build/html/_static/js/badge_only.js      |    1 +
 .../_static/js/html5shiv-printshiv.min.js     |    4 +
 docs/build/html/_static/js/html5shiv.min.js   |    4 +
 docs/build/html/_static/js/theme.js           |    1 +
 docs/build/html/_static/language_data.js      |  199 ++
 docs/build/html/_static/minus.png             |  Bin 0 -> 90 bytes
 docs/build/html/_static/plus.png              |  Bin 0 -> 90 bytes
 docs/build/html/_static/pygments.css          |   75 +
 docs/build/html/_static/searchtools.js        |  620 ++++
 docs/build/html/_static/sphinx_highlight.js   |  154 +
 docs/build/html/example.html                  |  220 ++
 docs/build/html/genindex.html                 |  220 ++
 docs/build/html/index.html                    |  193 ++
 docs/build/html/objects.inv                   |  Bin 0 -> 972 bytes
 docs/build/html/py-modindex.html              |  129 +
 docs/build/html/search.html                   |  126 +
 docs/build/html/searchindex.js                |    1 +
 docs/build/html/src.html                      |  552 ++++
 docs/make.bat                                 |   35 +
 docs/source/Figures/TernaryConcentrations.svg |  952 ++++++
 docs/source/Figures/TernaryPotential.svg      |  756 +++++
 docs/source/Figures/TernaryPressure.svg       |  679 +++++
 docs/source/conf.py                           |   46 +
 docs/source/example.rst                       |  105 +
 docs/source/index.rst                         |   82 +
 docs/source/src.rst                           |   20 +
 .../ReproducableCode/TernaryElectrolyte.py    |    2 +-
 examples/ReproducableCode/temp.py             |   66 +
 src/Eq02.py                                   |   15 +-
 src/Eq04.py                                   |    6 +-
 src/EqN.py                                    |    2 +
 .../ElectrolyticDiode.cpython-312.pyc         |  Bin 21332 -> 21327 bytes
 src/__pycache__/Eq02.cpython-312.pyc          |  Bin 16390 -> 16353 bytes
 src/__pycache__/Eq04.cpython-312.pyc          |  Bin 20398 -> 20383 bytes
 src/__pycache__/EqN.cpython-312.pyc           |  Bin 17913 -> 17841 bytes
 79 files changed, 12206 insertions(+), 9 deletions(-)
 create mode 100644 docs/Makefile
 create mode 100644 docs/build/doctrees/environment.pickle
 create mode 100644 docs/build/doctrees/example.doctree
 create mode 100644 docs/build/doctrees/index.doctree
 create mode 100644 docs/build/doctrees/src.doctree
 create mode 100644 docs/build/html/.buildinfo
 create mode 100644 docs/build/html/_images/TernaryConcentrations.svg
 create mode 100644 docs/build/html/_images/TernaryPotential.svg
 create mode 100644 docs/build/html/_images/TernaryPressure.svg
 create mode 100644 docs/build/html/_sources/example.rst.txt
 create mode 100644 docs/build/html/_sources/index.rst.txt
 create mode 100644 docs/build/html/_sources/src.rst.txt
 create mode 100644 docs/build/html/_static/_sphinx_javascript_frameworks_compat.js
 create mode 100644 docs/build/html/_static/basic.css
 create mode 100644 docs/build/html/_static/check-solid.svg
 create mode 100644 docs/build/html/_static/clipboard.min.js
 create mode 100644 docs/build/html/_static/copy-button.svg
 create mode 100644 docs/build/html/_static/copybutton.css
 create mode 100644 docs/build/html/_static/copybutton.js
 create mode 100644 docs/build/html/_static/copybutton_funcs.js
 create mode 100644 docs/build/html/_static/css/badge_only.css
 create mode 100644 docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff
 create mode 100644 docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff2
 create mode 100644 docs/build/html/_static/css/fonts/Roboto-Slab-Regular.woff
 create mode 100644 docs/build/html/_static/css/fonts/Roboto-Slab-Regular.woff2
 create mode 100644 docs/build/html/_static/css/fonts/fontawesome-webfont.eot
 create mode 100644 docs/build/html/_static/css/fonts/fontawesome-webfont.svg
 create mode 100644 docs/build/html/_static/css/fonts/fontawesome-webfont.ttf
 create mode 100644 docs/build/html/_static/css/fonts/fontawesome-webfont.woff
 create mode 100644 docs/build/html/_static/css/fonts/fontawesome-webfont.woff2
 create mode 100644 docs/build/html/_static/css/fonts/lato-bold-italic.woff
 create mode 100644 docs/build/html/_static/css/fonts/lato-bold-italic.woff2
 create mode 100644 docs/build/html/_static/css/fonts/lato-bold.woff
 create mode 100644 docs/build/html/_static/css/fonts/lato-bold.woff2
 create mode 100644 docs/build/html/_static/css/fonts/lato-normal-italic.woff
 create mode 100644 docs/build/html/_static/css/fonts/lato-normal-italic.woff2
 create mode 100644 docs/build/html/_static/css/fonts/lato-normal.woff
 create mode 100644 docs/build/html/_static/css/fonts/lato-normal.woff2
 create mode 100644 docs/build/html/_static/css/theme.css
 create mode 100644 docs/build/html/_static/doctools.js
 create mode 100644 docs/build/html/_static/documentation_options.js
 create mode 100644 docs/build/html/_static/file.png
 create mode 100644 docs/build/html/_static/jquery.js
 create mode 100644 docs/build/html/_static/js/badge_only.js
 create mode 100644 docs/build/html/_static/js/html5shiv-printshiv.min.js
 create mode 100644 docs/build/html/_static/js/html5shiv.min.js
 create mode 100644 docs/build/html/_static/js/theme.js
 create mode 100644 docs/build/html/_static/language_data.js
 create mode 100644 docs/build/html/_static/minus.png
 create mode 100644 docs/build/html/_static/plus.png
 create mode 100644 docs/build/html/_static/pygments.css
 create mode 100644 docs/build/html/_static/searchtools.js
 create mode 100644 docs/build/html/_static/sphinx_highlight.js
 create mode 100644 docs/build/html/example.html
 create mode 100644 docs/build/html/genindex.html
 create mode 100644 docs/build/html/index.html
 create mode 100644 docs/build/html/objects.inv
 create mode 100644 docs/build/html/py-modindex.html
 create mode 100644 docs/build/html/search.html
 create mode 100644 docs/build/html/searchindex.js
 create mode 100644 docs/build/html/src.html
 create mode 100644 docs/make.bat
 create mode 100644 docs/source/Figures/TernaryConcentrations.svg
 create mode 100644 docs/source/Figures/TernaryPotential.svg
 create mode 100644 docs/source/Figures/TernaryPressure.svg
 create mode 100644 docs/source/conf.py
 create mode 100644 docs/source/example.rst
 create mode 100644 docs/source/index.rst
 create mode 100644 docs/source/src.rst
 create mode 100644 examples/ReproducableCode/temp.py

diff --git a/README.md b/README.md
index 5984502..0564c32 100644
--- a/README.md
+++ b/README.md
@@ -8,7 +8,11 @@ Find the thesis at:
 
 ### Abstract
 
-- Insert Abstract from thesis
+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.
+
+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.
+
+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
 
diff --git a/docs/Makefile b/docs/Makefile
new file mode 100644
index 0000000..d0c3cbf
--- /dev/null
+++ b/docs/Makefile
@@ -0,0 +1,20 @@
+# Minimal makefile for Sphinx documentation
+#
+
+# You can set these variables from the command line, and also
+# from the environment for the first two.
+SPHINXOPTS    ?=
+SPHINXBUILD   ?= sphinx-build
+SOURCEDIR     = source
+BUILDDIR      = build
+
+# Put it first so that "make" without argument is like "make help".
+help:
+	@$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O)
+
+.PHONY: help Makefile
+
+# Catch-all target: route all unknown targets to Sphinx using the new
+# "make mode" option.  $(O) is meant as a shortcut for $(SPHINXOPTS).
+%: Makefile
+	@$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O)
diff --git a/docs/build/doctrees/environment.pickle b/docs/build/doctrees/environment.pickle
new file mode 100644
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diff --git a/docs/build/html/_sources/example.rst.txt b/docs/build/html/_sources/example.rst.txt
new file mode 100644
index 0000000..7fd5139
--- /dev/null
+++ b/docs/build/html/_sources/example.rst.txt
@@ -0,0 +1,105 @@
+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"
+
+| 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
+-----------------------------------------------------
+
+.. code-block:: python
+
+    import sys
+    import os
+
+    src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../..', 'src')
+    sys.path.insert(0, src_path)
+
+    from Eq04 import solve_System_4eq
+
+    del sys.path[0]
+
+    import matplotlib.pyplot as plt
+    import numpy as np
+
+Define parameters and boundary conditions
+-----------------------------------------
+
+.. code-block:: python
+
+    phi_left = 8.0
+    phi_right = 0.0
+    p_right = 0.0
+    y_A_R = 1/3
+    y_C_R = 1/3
+    z_A = -1.0
+    z_C = 1.0
+    K = 'incompressible'
+    Lambda2 = 8.553e-6
+    a2 = 7.5412e-4
+
+Define mesh and solver settings
+-------------------------------
+
+.. code-block:: python
+
+    number_cells = 1024
+    refinement_style = 'log'
+    rtol = 1e-8
+
+Solve the system
+----------------
+
+.. code-block:: python
+
+    y_A, y_C, phi, p, x = solve_System_4eq(phi_left, phi_right, p_right, z_A, z_C, y_A_R, y_C_R, K, Lambda2, a2, number_cells, relax_param=0.05, x0=0, x1=1,    refinement_style='hard_log', return_type='Vector', max_iter=1_000, rtol=rtol)
+
+Visualize the results
+---------------------
+
+.. code-block:: python
+
+    plt.figure()
+    plt.plot(x, phi)
+    plt.xlabel('x [-]')
+    plt.ylabel('$\\varphi$  [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.show()
+
+    plt.figure()
+    plt.plot(x, p)
+    plt.xlabel('x [-]')
+    plt.ylabel('$p$ [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.show()
+
+    plt.figure()
+    plt.plot(x, y_A, label='$y_A$')
+    plt.plot(x, y_C, label='$y_C$')
+    plt.plot(x, 1 - y_A - y_C, label='$y_S$')
+    plt.xlabel('x [-]')
+    plt.ylabel('$y_\\alpha$ [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.legend()
+    plt.show()
+
+The electric potential
+^^^^^^^^^^^^^^^^^^^^^^^
+
+.. image:: Figures/TernaryPotential.svg
+    :width: 600px
+
+The pressure 
+^^^^^^^^^^^^^
+
+.. image:: Figures/TernaryPressure.svg
+    :width: 600px
+
+The atomic fractions
+^^^^^^^^^^^^^^^^^^^^^
+.. image:: Figures/TernaryConcentrations.svg
+    :width: 600px
\ No newline at end of file
diff --git a/docs/build/html/_sources/index.rst.txt b/docs/build/html/_sources/index.rst.txt
new file mode 100644
index 0000000..0dc9c4b
--- /dev/null
+++ b/docs/build/html/_sources/index.rst.txt
@@ -0,0 +1,82 @@
+.. 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.
+
+Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation
+=====================================================================================================================
+
+.. admonition:: \ \
+
+   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: ToDo Add DOI
+
+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:
+
+- Insert link to thesis
+
+Abstract
+========
+
+| 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.
+
+| 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.
+
+| 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.
+
+.. 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
+
+Alternative installation
+========================
+| Use the "environment.yml" file to install all necessary environments
+
+.. code-block:: 
+
+   conda env create -f environment.yml
+
+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.
+
+Contact
+=======
+**Author**
+
+- Jan Habscheid
+- Jan.Habscheid@rwth-aachen.de
+
+**Supervisor**
+
+- Dr. Lambert Theissen
+- ACoM - Applied and Computational Mathematics
+- RWTH Aachen University
+- theisen@acom.rwth-aachen.de
+
+**Supervisor**
+
+- Prof. Dr. Manuel Torrilhon
+- ACoM - Applied and Computational Mathematics
+- RWTH Aachen University
+- mt@acom.rwth-aachen.de
+
+.. toctree::
+   src
+   example
+   :maxdepth: 2
+   :caption: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:
+   :hidden:
+
diff --git a/docs/build/html/_sources/src.rst.txt b/docs/build/html/_sources/src.rst.txt
new file mode 100644
index 0000000..f56f5d7
--- /dev/null
+++ b/docs/build/html/_sources/src.rst.txt
@@ -0,0 +1,20 @@
+Source Code
+===========
+
+.. 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
+
+.. automodule:: Helper_DoubleLayerCapacity
+    :members:
+    :undoc-members:
+    :show-inheritance:
\ No newline at end of file
diff --git a/docs/build/html/_static/_sphinx_javascript_frameworks_compat.js b/docs/build/html/_static/_sphinx_javascript_frameworks_compat.js
new file mode 100644
index 0000000..8141580
--- /dev/null
+++ b/docs/build/html/_static/_sphinx_javascript_frameworks_compat.js
@@ -0,0 +1,123 @@
+/* Compatability shim for jQuery and underscores.js.
+ *
+ * Copyright Sphinx contributors
+ * Released under the two clause BSD licence
+ */
+
+/**
+ * small helper function to urldecode strings
+ *
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL
+ */
+jQuery.urldecode = function(x) {
+    if (!x) {
+        return x
+    }
+    return decodeURIComponent(x.replace(/\+/g, ' '));
+};
+
+/**
+ * small helper function to urlencode strings
+ */
+jQuery.urlencode = encodeURIComponent;
+
+/**
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+        var key = jQuery.urldecode(tmp[0]);
+        var value = jQuery.urldecode(tmp[1]);
+        if (key in result)
+            result[key].push(value);
+        else
+            result[key] = [value];
+    }
+    return result;
+};
+
+/**
+ * highlight a given string on a jquery object by wrapping it in
+ * span elements with the given class name.
+ */
+jQuery.fn.highlightText = function(text, className) {
+    function highlight(node, addItems) {
+        if (node.nodeType === 3) {
+            var val = node.nodeValue;
+            var pos = val.toLowerCase().indexOf(text);
+            if (pos >= 0 &&
+                !jQuery(node.parentNode).hasClass(className) &&
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+                var span;
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+                highlight(this, addItems);
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+    var addItems = [];
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+        highlight(this, addItems);
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+    for (var i = 0; i < addItems.length; ++i) {
+        jQuery(addItems[i].parent).before(addItems[i].target);
+    }
+    return result;
+};
+
+/*
+ * backward compatibility for jQuery.browser
+ * This will be supported until firefox bug is fixed.
+ */
+if (!jQuery.browser) {
+    jQuery.uaMatch = function(ua) {
+        ua = ua.toLowerCase();
+
+        var match = /(chrome)[ \/]([\w.]+)/.exec(ua) ||
+            /(webkit)[ \/]([\w.]+)/.exec(ua) ||
+            /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) ||
+            /(msie) ([\w.]+)/.exec(ua) ||
+            ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) ||
+            [];
+
+        return {
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+            version: match[ 2 ] || "0"
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+    };
+    jQuery.browser = {};
+    jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true;
+}
diff --git a/docs/build/html/_static/basic.css b/docs/build/html/_static/basic.css
new file mode 100644
index 0000000..f316efc
--- /dev/null
+++ b/docs/build/html/_static/basic.css
@@ -0,0 +1,925 @@
+/*
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+ * ~~~~~~~~~
+ *
+ * Sphinx stylesheet -- basic theme.
+ *
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+
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+
+div.section::after {
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+ol.upperalpha {
+    list-style: upper-alpha;
+}
+
+ol.lowerroman {
+    list-style: lower-roman;
+}
+
+ol.upperroman {
+    list-style: upper-roman;
+}
+
+:not(li) > ol > li:first-child > :first-child,
+:not(li) > ul > li:first-child > :first-child {
+    margin-top: 0px;
+}
+
+:not(li) > ol > li:last-child > :last-child,
+:not(li) > ul > li:last-child > :last-child {
+    margin-bottom: 0px;
+}
+
+ol.simple ol p,
+ol.simple ul p,
+ul.simple ol p,
+ul.simple ul p {
+    margin-top: 0;
+}
+
+ol.simple > li:not(:first-child) > p,
+ul.simple > li:not(:first-child) > p {
+    margin-top: 0;
+}
+
+ol.simple p,
+ul.simple p {
+    margin-bottom: 0;
+}
+
+aside.footnote > span,
+div.citation > span {
+    float: left;
+}
+aside.footnote > span:last-of-type,
+div.citation > span:last-of-type {
+  padding-right: 0.5em;
+}
+aside.footnote > p {
+  margin-left: 2em;
+}
+div.citation > p {
+  margin-left: 4em;
+}
+aside.footnote > p:last-of-type,
+div.citation > p:last-of-type {
+    margin-bottom: 0em;
+}
+aside.footnote > p:last-of-type:after,
+div.citation > p:last-of-type:after {
+    content: "";
+    clear: both;
+}
+
+dl.field-list {
+    display: grid;
+    grid-template-columns: fit-content(30%) auto;
+}
+
+dl.field-list > dt {
+    font-weight: bold;
+    word-break: break-word;
+    padding-left: 0.5em;
+    padding-right: 5px;
+}
+
+dl.field-list > dd {
+    padding-left: 0.5em;
+    margin-top: 0em;
+    margin-left: 0em;
+    margin-bottom: 0em;
+}
+
+dl {
+    margin-bottom: 15px;
+}
+
+dd > :first-child {
+    margin-top: 0px;
+}
+
+dd ul, dd table {
+    margin-bottom: 10px;
+}
+
+dd {
+    margin-top: 3px;
+    margin-bottom: 10px;
+    margin-left: 30px;
+}
+
+.sig dd {
+    margin-top: 0px;
+    margin-bottom: 0px;
+}
+
+.sig dl {
+    margin-top: 0px;
+    margin-bottom: 0px;
+}
+
+dl > dd:last-child,
+dl > dd:last-child > :last-child {
+    margin-bottom: 0;
+}
+
+dt:target, span.highlighted {
+    background-color: #fbe54e;
+}
+
+rect.highlighted {
+    fill: #fbe54e;
+}
+
+dl.glossary dt {
+    font-weight: bold;
+    font-size: 1.1em;
+}
+
+.versionmodified {
+    font-style: italic;
+}
+
+.system-message {
+    background-color: #fda;
+    padding: 5px;
+    border: 3px solid red;
+}
+
+.footnote:target  {
+    background-color: #ffa;
+}
+
+.line-block {
+    display: block;
+    margin-top: 1em;
+    margin-bottom: 1em;
+}
+
+.line-block .line-block {
+    margin-top: 0;
+    margin-bottom: 0;
+    margin-left: 1.5em;
+}
+
+.guilabel, .menuselection {
+    font-family: sans-serif;
+}
+
+.accelerator {
+    text-decoration: underline;
+}
+
+.classifier {
+    font-style: oblique;
+}
+
+.classifier:before {
+    font-style: normal;
+    margin: 0 0.5em;
+    content: ":";
+    display: inline-block;
+}
+
+abbr, acronym {
+    border-bottom: dotted 1px;
+    cursor: help;
+}
+
+.translated {
+    background-color: rgba(207, 255, 207, 0.2)
+}
+
+.untranslated {
+    background-color: rgba(255, 207, 207, 0.2)
+}
+
+/* -- code displays --------------------------------------------------------- */
+
+pre {
+    overflow: auto;
+    overflow-y: hidden;  /* fixes display issues on Chrome browsers */
+}
+
+pre, div[class*="highlight-"] {
+    clear: both;
+}
+
+span.pre {
+    -moz-hyphens: none;
+    -ms-hyphens: none;
+    -webkit-hyphens: none;
+    hyphens: none;
+    white-space: nowrap;
+}
+
+div[class*="highlight-"] {
+    margin: 1em 0;
+}
+
+td.linenos pre {
+    border: 0;
+    background-color: transparent;
+    color: #aaa;
+}
+
+table.highlighttable {
+    display: block;
+}
+
+table.highlighttable tbody {
+    display: block;
+}
+
+table.highlighttable tr {
+    display: flex;
+}
+
+table.highlighttable td {
+    margin: 0;
+    padding: 0;
+}
+
+table.highlighttable td.linenos {
+    padding-right: 0.5em;
+}
+
+table.highlighttable td.code {
+    flex: 1;
+    overflow: hidden;
+}
+
+.highlight .hll {
+    display: block;
+}
+
+div.highlight pre,
+table.highlighttable pre {
+    margin: 0;
+}
+
+div.code-block-caption + div {
+    margin-top: 0;
+}
+
+div.code-block-caption {
+    margin-top: 1em;
+    padding: 2px 5px;
+    font-size: small;
+}
+
+div.code-block-caption code {
+    background-color: transparent;
+}
+
+table.highlighttable td.linenos,
+span.linenos,
+div.highlight span.gp {  /* gp: Generic.Prompt */
+  user-select: none;
+  -webkit-user-select: text; /* Safari fallback only */
+  -webkit-user-select: none; /* Chrome/Safari */
+  -moz-user-select: none; /* Firefox */
+  -ms-user-select: none; /* IE10+ */
+}
+
+div.code-block-caption span.caption-number {
+    padding: 0.1em 0.3em;
+    font-style: italic;
+}
+
+div.code-block-caption span.caption-text {
+}
+
+div.literal-block-wrapper {
+    margin: 1em 0;
+}
+
+code.xref, a code {
+    background-color: transparent;
+    font-weight: bold;
+}
+
+h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
+    background-color: transparent;
+}
+
+.viewcode-link {
+    float: right;
+}
+
+.viewcode-back {
+    float: right;
+    font-family: sans-serif;
+}
+
+div.viewcode-block:target {
+    margin: -1px -10px;
+    padding: 0 10px;
+}
+
+/* -- math display ---------------------------------------------------------- */
+
+img.math {
+    vertical-align: middle;
+}
+
+div.body div.math p {
+    text-align: center;
+}
+
+span.eqno {
+    float: right;
+}
+
+span.eqno a.headerlink {
+    position: absolute;
+    z-index: 1;
+}
+
+div.math:hover a.headerlink {
+    visibility: visible;
+}
+
+/* -- printout stylesheet --------------------------------------------------- */
+
+@media print {
+    div.document,
+    div.documentwrapper,
+    div.bodywrapper {
+        margin: 0 !important;
+        width: 100%;
+    }
+
+    div.sphinxsidebar,
+    div.related,
+    div.footer,
+    #top-link {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/docs/build/html/_static/check-solid.svg b/docs/build/html/_static/check-solid.svg
new file mode 100644
index 0000000..92fad4b
--- /dev/null
+++ b/docs/build/html/_static/check-solid.svg
@@ -0,0 +1,4 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>
diff --git a/docs/build/html/_static/clipboard.min.js b/docs/build/html/_static/clipboard.min.js
new file mode 100644
index 0000000..54b3c46
--- /dev/null
+++ b/docs/build/html/_static/clipboard.min.js
@@ -0,0 +1,7 @@
+/*!
+ * clipboard.js v2.0.8
+ * https://clipboardjs.com/
+ *
+ * Licensed MIT © Zeno Rocha
+ */
+!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return o}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),c=n.n(e);function a(t){try{return document.execCommand(t)}catch(t){return}}var f=function(t){t=c()(t);return a("cut"),t};var l=function(t){var e,n,o,r=1<arguments.length&&void 0!==arguments[1]?arguments[1]:{container:document.body},i="";return"string"==typeof t?(e=t,n="rtl"===document.documentElement.getAttribute("dir"),(o=document.createElement("textarea")).style.fontSize="12pt",o.style.border="0",o.style.padding="0",o.style.margin="0",o.style.position="absolute",o.style[n?"right":"left"]="-9999px",n=window.pageYOffset||document.documentElement.scrollTop,o.style.top="".concat(n,"px"),o.setAttribute("readonly",""),o.value=e,o=o,r.container.appendChild(o),i=c()(o),a("copy"),o.remove()):(i=c()(t),a("copy")),i};function r(t){return(r="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}var s=function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{},e=t.action,n=void 0===e?"copy":e,o=t.container,e=t.target,t=t.text;if("copy"!==n&&"cut"!==n)throw new Error('Invalid "action" value, use either "copy" or "cut"');if(void 0!==e){if(!e||"object"!==r(e)||1!==e.nodeType)throw new Error('Invalid "target" value, use a valid Element');if("copy"===n&&e.hasAttribute("disabled"))throw new Error('Invalid "target" attribute. 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\ No newline at end of file
diff --git a/docs/build/html/_static/copy-button.svg b/docs/build/html/_static/copy-button.svg
new file mode 100644
index 0000000..9c074da
--- /dev/null
+++ b/docs/build/html/_static/copy-button.svg
@@ -0,0 +1,5 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>
diff --git a/docs/build/html/_static/copybutton.css b/docs/build/html/_static/copybutton.css
new file mode 100644
index 0000000..f1916ec
--- /dev/null
+++ b/docs/build/html/_static/copybutton.css
@@ -0,0 +1,94 @@
+/* Copy buttons */
+button.copybtn {
+    position: absolute;
+    display: flex;
+    top: .3em;
+    right: .3em;
+    width: 1.7em;
+    height: 1.7em;
+	opacity: 0;
+    transition: opacity 0.3s, border .3s, background-color .3s;
+    user-select: none;
+    padding: 0;
+    border: none;
+    outline: none;
+    border-radius: 0.4em;
+    /* The colors that GitHub uses */
+    border: #1b1f2426 1px solid;
+    background-color: #f6f8fa;
+    color: #57606a;
+}
+
+button.copybtn.success {
+    border-color: #22863a;
+    color: #22863a;
+}
+
+button.copybtn svg {
+    stroke: currentColor;
+    width: 1.5em;
+    height: 1.5em;
+    padding: 0.1em;
+}
+
+div.highlight  {
+    position: relative;
+}
+
+/* Show the copybutton */
+.highlight:hover button.copybtn, button.copybtn.success {
+	opacity: 1;
+}
+
+.highlight button.copybtn:hover {
+    background-color: rgb(235, 235, 235);
+}
+
+.highlight button.copybtn:active {
+    background-color: rgb(187, 187, 187);
+}
+
+/**
+ * A minimal CSS-only tooltip copied from:
+ *   https://codepen.io/mildrenben/pen/rVBrpK
+ *
+ * To use, write HTML like the following:
+ *
+ * <p class="o-tooltip--left" data-tooltip="Hey">Short</p>
+ */
+ .o-tooltip--left {
+  position: relative;
+ }
+
+ .o-tooltip--left:after {
+    opacity: 0;
+    visibility: hidden;
+    position: absolute;
+    content: attr(data-tooltip);
+    padding: .2em;
+    font-size: .8em;
+    left: -.2em;
+    background: grey;
+    color: white;
+    white-space: nowrap;
+    z-index: 2;
+    border-radius: 2px;
+    transform: translateX(-102%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+}
+
+.o-tooltip--left:hover:after {
+    display: block;
+    opacity: 1;
+    visibility: visible;
+    transform: translateX(-100%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+    transition-delay: .5s;
+}
+
+/* By default the copy button shouldn't show up when printing a page */
+@media print {
+    button.copybtn {
+        display: none;
+    }
+}
diff --git a/docs/build/html/_static/copybutton.js b/docs/build/html/_static/copybutton.js
new file mode 100644
index 0000000..2ea7ff3
--- /dev/null
+++ b/docs/build/html/_static/copybutton.js
@@ -0,0 +1,248 @@
+// Localization support
+const messages = {
+  'en': {
+    'copy': 'Copy',
+    'copy_to_clipboard': 'Copy to clipboard',
+    'copy_success': 'Copied!',
+    'copy_failure': 'Failed to copy',
+  },
+  'es' : {
+    'copy': 'Copiar',
+    'copy_to_clipboard': 'Copiar al portapapeles',
+    'copy_success': '¡Copiado!',
+    'copy_failure': 'Error al copiar',
+  },
+  'de' : {
+    'copy': 'Kopieren',
+    'copy_to_clipboard': 'In die Zwischenablage kopieren',
+    'copy_success': 'Kopiert!',
+    'copy_failure': 'Fehler beim Kopieren',
+  },
+  'fr' : {
+    'copy': 'Copier',
+    'copy_to_clipboard': 'Copier dans le presse-papier',
+    'copy_success': 'Copié !',
+    'copy_failure': 'Échec de la copie',
+  },
+  'ru': {
+    'copy': 'Скопировать',
+    'copy_to_clipboard': 'Скопировать в буфер',
+    'copy_success': 'Скопировано!',
+    'copy_failure': 'Не удалось скопировать',
+  },
+  'zh-CN': {
+    'copy': '复制',
+    'copy_to_clipboard': '复制到剪贴板',
+    'copy_success': '复制成功!',
+    'copy_failure': '复制失败',
+  },
+  'it' : {
+    'copy': 'Copiare',
+    'copy_to_clipboard': 'Copiato negli appunti',
+    'copy_success': 'Copiato!',
+    'copy_failure': 'Errore durante la copia',
+  }
+}
+
+let locale = 'en'
+if( document.documentElement.lang !== undefined
+    && messages[document.documentElement.lang] !== undefined ) {
+  locale = document.documentElement.lang
+}
+
+let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT;
+if (doc_url_root == '#') {
+    doc_url_root = '';
+}
+
+/**
+ * SVG files for our copy buttons
+ */
+let iconCheck = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_success']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>`
+
+// If the user specified their own SVG use that, otherwise use the default
+let iconCopy = ``;
+if (!iconCopy) {
+  iconCopy = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_to_clipboard']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>`
+}
+
+/**
+ * Set up copy/paste for code blocks
+ */
+
+const runWhenDOMLoaded = cb => {
+  if (document.readyState != 'loading') {
+    cb()
+  } else if (document.addEventListener) {
+    document.addEventListener('DOMContentLoaded', cb)
+  } else {
+    document.attachEvent('onreadystatechange', function() {
+      if (document.readyState == 'complete') cb()
+    })
+  }
+}
+
+const codeCellId = index => `codecell${index}`
+
+// Clears selected text since ClipboardJS will select the text when copying
+const clearSelection = () => {
+  if (window.getSelection) {
+    window.getSelection().removeAllRanges()
+  } else if (document.selection) {
+    document.selection.empty()
+  }
+}
+
+// Changes tooltip text for a moment, then changes it back
+// We want the timeout of our `success` class to be a bit shorter than the
+// tooltip and icon change, so that we can hide the icon before changing back.
+var timeoutIcon = 2000;
+var timeoutSuccessClass = 1500;
+
+const temporarilyChangeTooltip = (el, oldText, newText) => {
+  el.setAttribute('data-tooltip', newText)
+  el.classList.add('success')
+  // Remove success a little bit sooner than we change the tooltip
+  // So that we can use CSS to hide the copybutton first
+  setTimeout(() => el.classList.remove('success'), timeoutSuccessClass)
+  setTimeout(() => el.setAttribute('data-tooltip', oldText), timeoutIcon)
+}
+
+// Changes the copy button icon for two seconds, then changes it back
+const temporarilyChangeIcon = (el) => {
+  el.innerHTML = iconCheck;
+  setTimeout(() => {el.innerHTML = iconCopy}, timeoutIcon)
+}
+
+const addCopyButtonToCodeCells = () => {
+  // If ClipboardJS hasn't loaded, wait a bit and try again. This
+  // happens because we load ClipboardJS asynchronously.
+  if (window.ClipboardJS === undefined) {
+    setTimeout(addCopyButtonToCodeCells, 250)
+    return
+  }
+
+  // Add copybuttons to all of our code cells
+  const COPYBUTTON_SELECTOR = 'div.highlight pre';
+  const codeCells = document.querySelectorAll(COPYBUTTON_SELECTOR)
+  codeCells.forEach((codeCell, index) => {
+    const id = codeCellId(index)
+    codeCell.setAttribute('id', id)
+
+    const clipboardButton = id =>
+    `<button class="copybtn o-tooltip--left" data-tooltip="${messages[locale]['copy']}" data-clipboard-target="#${id}">
+      ${iconCopy}
+    </button>`
+    codeCell.insertAdjacentHTML('afterend', clipboardButton(id))
+  })
+
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
+
+
+var copyTargetText = (trigger) => {
+  var target = document.querySelector(trigger.attributes['data-clipboard-target'].value);
+
+  // get filtered text
+  let exclude = '.linenos';
+
+  let text = filterText(target, exclude);
+  return formatCopyText(text, '', false, true, true, true, '', '')
+}
+
+  // Initialize with a callback so we can modify the text before copy
+  const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText})
+
+  // Update UI with error/success messages
+  clipboard.on('success', event => {
+    clearSelection()
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success'])
+    temporarilyChangeIcon(event.trigger)
+  })
+
+  clipboard.on('error', event => {
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure'])
+  })
+}
+
+runWhenDOMLoaded(addCopyButtonToCodeCells)
\ No newline at end of file
diff --git a/docs/build/html/_static/copybutton_funcs.js b/docs/build/html/_static/copybutton_funcs.js
new file mode 100644
index 0000000..dbe1aaa
--- /dev/null
+++ b/docs/build/html/_static/copybutton_funcs.js
@@ -0,0 +1,73 @@
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+export function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
diff --git a/docs/build/html/_static/css/badge_only.css b/docs/build/html/_static/css/badge_only.css
new file mode 100644
index 0000000..c718cee
--- /dev/null
+++ b/docs/build/html/_static/css/badge_only.css
@@ -0,0 +1 @@
+.clearfix{*zoom:1}.clearfix:after,.clearfix:before{display:table;content:""}.clearfix:after{clear:both}@font-face{font-family:FontAwesome;font-style:normal;font-weight:400;src:url(fonts/fontawesome-webfont.eot?674f50d287a8c48dc19ba404d20fe713?#iefix) format("embedded-opentype"),url(fonts/fontawesome-webfont.woff2?af7ae505a9eed503f8b8e6982036873e) format("woff2"),url(fonts/fontawesome-webfont.woff?fee66e712a8a08eef5805a46892932ad) format("woff"),url(fonts/fontawesome-webfont.ttf?b06871f281fee6b241d60582ae9369b9) format("truetype"),url(fonts/fontawesome-webfont.svg?912ec66d7572ff821749319396470bde#FontAwesome) format("svg")}.fa:before{font-family:FontAwesome;font-style:normal;font-weight:400;line-height:1}.fa:before,a .fa{text-decoration:inherit}.fa:before,a .fa,li .fa{display:inline-block}li .fa-large:before{width:1.875em}ul.fas{list-style-type:none;margin-left:2em;text-indent:-.8em}ul.fas li .fa{width:.8em}ul.fas li .fa-large:before{vertical-align:baseline}.fa-book:before,.icon-book:before{content:"\f02d"}.fa-caret-down:before,.icon-caret-down:before{content:"\f0d7"}.fa-caret-up:before,.icon-caret-up:before{content:"\f0d8"}.fa-caret-left:before,.icon-caret-left:before{content:"\f0d9"}.fa-caret-right:before,.icon-caret-right:before{content:"\f0da"}.rst-versions{position:fixed;bottom:0;left:0;width:300px;color:#fcfcfc;background:#1f1d1d;font-family:Lato,proxima-nova,Helvetica Neue,Arial,sans-serif;z-index:400}.rst-versions a{color:#2980b9;text-decoration:none}.rst-versions .rst-badge-small{display:none}.rst-versions .rst-current-version{padding:12px;background-color:#272525;display:block;text-align:right;font-size:90%;cursor:pointer;color:#27ae60}.rst-versions .rst-current-version:after{clear:both;content:"";display:block}.rst-versions .rst-current-version .fa{color:#fcfcfc}.rst-versions .rst-current-version .fa-book,.rst-versions .rst-current-version .icon-book{float:left}.rst-versions .rst-current-version.rst-out-of-date{background-color:#e74c3c;color:#fff}.rst-versions .rst-current-version.rst-active-old-version{background-color:#f1c40f;color:#000}.rst-versions.shift-up{height:auto;max-height:100%;overflow-y:scroll}.rst-versions.shift-up .rst-other-versions{display:block}.rst-versions .rst-other-versions{font-size:90%;padding:12px;color:grey;display:none}.rst-versions .rst-other-versions hr{display:block;height:1px;border:0;margin:20px 0;padding:0;border-top:1px solid #413d3d}.rst-versions .rst-other-versions dd{display:inline-block;margin:0}.rst-versions .rst-other-versions dd a{display:inline-block;padding:6px;color:#fcfcfc}.rst-versions.rst-badge{width:auto;bottom:20px;right:20px;left:auto;border:none;max-width:300px;max-height:90%}.rst-versions.rst-badge .fa-book,.rst-versions.rst-badge .icon-book{float:none;line-height:30px}.rst-versions.rst-badge.shift-up .rst-current-version{text-align:right}.rst-versions.rst-badge.shift-up .rst-current-version .fa-book,.rst-versions.rst-badge.shift-up .rst-current-version .icon-book{float:left}.rst-versions.rst-badge>.rst-current-version{width:auto;height:30px;line-height:30px;padding:0 6px;display:block;text-align:center}@media screen and (max-width:768px){.rst-versions{width:85%;display:none}.rst-versions.shift{display:block}}
\ No newline at end of file
diff --git a/docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff b/docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff
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diff --git a/docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff2 b/docs/build/html/_static/css/fonts/Roboto-Slab-Bold.woff2
new file mode 100644
index 0000000000000000000000000000000000000000..7059e23142aae3d8bad6067fc734a6cffec779c9
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new file mode 100644
index 0000000..855c845
--- /dev/null
+++ b/docs/build/html/_static/css/fonts/fontawesome-webfont.svg
@@ -0,0 +1,2671 @@
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+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd" >
+<svg>
+<metadata>
+Created by FontForge 20120731 at Mon Oct 24 17:37:40 2016
+ By ,,,
+Copyright Dave Gandy 2016. All rights reserved.
+</metadata>
+<defs>
+<font id="FontAwesome" horiz-adv-x="1536" >
+  <font-face 
+    font-family="FontAwesome"
+    font-weight="400"
+    font-stretch="normal"
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+    underline-thickness="0"
+    underline-position="0"
+    unicode-range="U+0020-F500"
+  />
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+d="M224 112h448v1312h-448v-1312zM112 0v1536h672v-1536h-672z" />
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+    <glyph glyph-name="nonmarkingreturn" horiz-adv-x="597" 
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+    <glyph glyph-name="space" unicode=" " horiz-adv-x="448" 
+ />
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+ />
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+ />
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diff --git a/docs/build/html/_static/css/fonts/fontawesome-webfont.woff2 b/docs/build/html/_static/css/fonts/fontawesome-webfont.woff2
new file mode 100644
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diff --git a/docs/build/html/_static/css/fonts/lato-bold-italic.woff b/docs/build/html/_static/css/fonts/lato-bold-italic.woff
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diff --git a/docs/build/html/_static/doctools.js b/docs/build/html/_static/doctools.js
new file mode 100644
index 0000000..4d67807
--- /dev/null
+++ b/docs/build/html/_static/doctools.js
@@ -0,0 +1,156 @@
+/*
+ * doctools.js
+ * ~~~~~~~~~~~
+ *
+ * Base JavaScript utilities for all Sphinx HTML documentation.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+const BLACKLISTED_KEY_CONTROL_ELEMENTS = new Set([
+  "TEXTAREA",
+  "INPUT",
+  "SELECT",
+  "BUTTON",
+]);
+
+const _ready = (callback) => {
+  if (document.readyState !== "loading") {
+    callback();
+  } else {
+    document.addEventListener("DOMContentLoaded", callback);
+  }
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const Documentation = {
+  init: () => {
+    Documentation.initDomainIndexTable();
+    Documentation.initOnKeyListeners();
+  },
+
+  /**
+   * i18n support
+   */
+  TRANSLATIONS: {},
+  PLURAL_EXPR: (n) => (n === 1 ? 0 : 1),
+  LOCALE: "unknown",
+
+  // gettext and ngettext don't access this so that the functions
+  // can safely bound to a different name (_ = Documentation.gettext)
+  gettext: (string) => {
+    const translated = Documentation.TRANSLATIONS[string];
+    switch (typeof translated) {
+      case "undefined":
+        return string; // no translation
+      case "string":
+        return translated; // translation exists
+      default:
+        return translated[0]; // (singular, plural) translation tuple exists
+    }
+  },
+
+  ngettext: (singular, plural, n) => {
+    const translated = Documentation.TRANSLATIONS[singular];
+    if (typeof translated !== "undefined")
+      return translated[Documentation.PLURAL_EXPR(n)];
+    return n === 1 ? singular : plural;
+  },
+
+  addTranslations: (catalog) => {
+    Object.assign(Documentation.TRANSLATIONS, catalog.messages);
+    Documentation.PLURAL_EXPR = new Function(
+      "n",
+      `return (${catalog.plural_expr})`
+    );
+    Documentation.LOCALE = catalog.locale;
+  },
+
+  /**
+   * helper function to focus on search bar
+   */
+  focusSearchBar: () => {
+    document.querySelectorAll("input[name=q]")[0]?.focus();
+  },
+
+  /**
+   * Initialise the domain index toggle buttons
+   */
+  initDomainIndexTable: () => {
+    const toggler = (el) => {
+      const idNumber = el.id.substr(7);
+      const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`);
+      if (el.src.substr(-9) === "minus.png") {
+        el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`;
+        toggledRows.forEach((el) => (el.style.display = "none"));
+      } else {
+        el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`;
+        toggledRows.forEach((el) => (el.style.display = ""));
+      }
+    };
+
+    const togglerElements = document.querySelectorAll("img.toggler");
+    togglerElements.forEach((el) =>
+      el.addEventListener("click", (event) => toggler(event.currentTarget))
+    );
+    togglerElements.forEach((el) => (el.style.display = ""));
+    if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler);
+  },
+
+  initOnKeyListeners: () => {
+    // only install a listener if it is really needed
+    if (
+      !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS &&
+      !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS
+    )
+      return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.altKey || event.ctrlKey || event.metaKey) return;
+
+      if (!event.shiftKey) {
+        switch (event.key) {
+          case "ArrowLeft":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const prevLink = document.querySelector('link[rel="prev"]');
+            if (prevLink && prevLink.href) {
+              window.location.href = prevLink.href;
+              event.preventDefault();
+            }
+            break;
+          case "ArrowRight":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const nextLink = document.querySelector('link[rel="next"]');
+            if (nextLink && nextLink.href) {
+              window.location.href = nextLink.href;
+              event.preventDefault();
+            }
+            break;
+        }
+      }
+
+      // some keyboard layouts may need Shift to get /
+      switch (event.key) {
+        case "/":
+          if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break;
+          Documentation.focusSearchBar();
+          event.preventDefault();
+      }
+    });
+  },
+};
+
+// quick alias for translations
+const _ = Documentation.gettext;
+
+_ready(Documentation.init);
diff --git a/docs/build/html/_static/documentation_options.js b/docs/build/html/_static/documentation_options.js
new file mode 100644
index 0000000..2efaa85
--- /dev/null
+++ b/docs/build/html/_static/documentation_options.js
@@ -0,0 +1,13 @@
+const DOCUMENTATION_OPTIONS = {
+    VERSION: 'September, 2024',
+    LANGUAGE: 'en',
+    COLLAPSE_INDEX: false,
+    BUILDER: 'html',
+    FILE_SUFFIX: '.html',
+    LINK_SUFFIX: '.html',
+    HAS_SOURCE: true,
+    SOURCELINK_SUFFIX: '.txt',
+    NAVIGATION_WITH_KEYS: false,
+    SHOW_SEARCH_SUMMARY: true,
+    ENABLE_SEARCH_SHORTCUTS: true,
+};
\ No newline at end of file
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diff --git a/docs/build/html/_static/jquery.js b/docs/build/html/_static/jquery.js
new file mode 100644
index 0000000..c4c6022
--- /dev/null
+++ b/docs/build/html/_static/jquery.js
@@ -0,0 +1,2 @@
+/*! jQuery v3.6.0 | (c) OpenJS Foundation and other contributors | jquery.org/license */
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+!function(n){var e={};function t(i){if(e[i])return e[i].exports;var o=e[i]={i:i,l:!1,exports:{}};return n[i].call(o.exports,o,o.exports,t),o.l=!0,o.exports}t.m=n,t.c=e,t.d=function(n,e,i){t.o(n,e)||Object.defineProperty(n,e,{enumerable:!0,get:i})},t.r=function(n){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(n,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(n,"__esModule",{value:!0})},t.t=function(n,e){if(1&e&&(n=t(n)),8&e)return n;if(4&e&&"object"==typeof n&&n&&n.__esModule)return n;var i=Object.create(null);if(t.r(i),Object.defineProperty(i,"default",{enumerable:!0,value:n}),2&e&&"string"!=typeof n)for(var o in n)t.d(i,o,function(e){return n[e]}.bind(null,o));return i},t.n=function(n){var e=n&&n.__esModule?function(){return n.default}:function(){return n};return t.d(e,"a",e),e},t.o=function(n,e){return Object.prototype.hasOwnProperty.call(n,e)},t.p="",t(t.s=0)}([function(n,e,t){t(1),n.exports=t(3)},function(n,e,t){(function(){var e="undefined"!=typeof window?window.jQuery:t(2);n.exports.ThemeNav={navBar:null,win:null,winScroll:!1,winResize:!1,linkScroll:!1,winPosition:0,winHeight:null,docHeight:null,isRunning:!1,enable:function(n){var t=this;void 0===n&&(n=!0),t.isRunning||(t.isRunning=!0,e((function(e){t.init(e),t.reset(),t.win.on("hashchange",t.reset),n&&t.win.on("scroll",(function(){t.linkScroll||t.winScroll||(t.winScroll=!0,requestAnimationFrame((function(){t.onScroll()})))})),t.win.on("resize",(function(){t.winResize||(t.winResize=!0,requestAnimationFrame((function(){t.onResize()})))})),t.onResize()})))},enableSticky:function(){this.enable(!0)},init:function(n){n(document);var e=this;this.navBar=n("div.wy-side-scroll:first"),this.win=n(window),n(document).on("click","[data-toggle='wy-nav-top']",(function(){n("[data-toggle='wy-nav-shift']").toggleClass("shift"),n("[data-toggle='rst-versions']").toggleClass("shift")})).on("click",".wy-menu-vertical .current ul li a",(function(){var 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[id="'+n.substring(1)+'"]').closest("div.section");0===(t=e.find('[href="#'+i.attr("id")+'"]')).length&&(t=e.find('[href="#"]'))}if(t.length>0){$(".wy-menu-vertical .current").removeClass("current").attr("aria-expanded","false"),t.addClass("current").attr("aria-expanded","true"),t.closest("li.toctree-l1").parent().addClass("current").attr("aria-expanded","true");for(let n=1;n<=10;n++)t.closest("li.toctree-l"+n).addClass("current").attr("aria-expanded","true");t[0].scrollIntoView()}}catch(n){console.log("Error expanding nav for anchor",n)}},onScroll:function(){this.winScroll=!1;var n=this.win.scrollTop(),e=n+this.winHeight,t=this.navBar.scrollTop()+(n-this.winPosition);n<0||e>this.docHeight||(this.navBar.scrollTop(t),this.winPosition=n)},onResize:function(){this.winResize=!1,this.winHeight=this.win.height(),this.docHeight=$(document).height()},hashChange:function(){this.linkScroll=!0,this.win.one("hashchange",(function(){this.linkScroll=!1}))},toggleCurrent:function(n){var e=n.closest("li");e.siblings("li.current").removeClass("current").attr("aria-expanded","false"),e.siblings().find("li.current").removeClass("current").attr("aria-expanded","false");var t=e.find("> ul li");t.length&&(t.removeClass("current").attr("aria-expanded","false"),e.toggleClass("current").attr("aria-expanded",(function(n,e){return"true"==e?"false":"true"})))}},"undefined"!=typeof window&&(window.SphinxRtdTheme={Navigation:n.exports.ThemeNav,StickyNav:n.exports.ThemeNav}),function(){for(var n=0,e=["ms","moz","webkit","o"],t=0;t<e.length&&!window.requestAnimationFrame;++t)window.requestAnimationFrame=window[e[t]+"RequestAnimationFrame"],window.cancelAnimationFrame=window[e[t]+"CancelAnimationFrame"]||window[e[t]+"CancelRequestAnimationFrame"];window.requestAnimationFrame||(window.requestAnimationFrame=function(e,t){var i=(new Date).getTime(),o=Math.max(0,16-(i-n)),r=window.setTimeout((function(){e(i+o)}),o);return 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\ No newline at end of file
diff --git a/docs/build/html/_static/language_data.js b/docs/build/html/_static/language_data.js
new file mode 100644
index 0000000..367b8ed
--- /dev/null
+++ b/docs/build/html/_static/language_data.js
@@ -0,0 +1,199 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
+
+
+/* Non-minified version is copied as a separate JS file, if available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
diff --git a/docs/build/html/_static/minus.png b/docs/build/html/_static/minus.png
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GIT binary patch
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literal 0
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diff --git a/docs/build/html/_static/plus.png b/docs/build/html/_static/plus.png
new file mode 100644
index 0000000000000000000000000000000000000000..7107cec93a979b9a5f64843235a16651d563ce2d
GIT binary patch
literal 90
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pT5Cj47Kj#R>InVM$K{a2!oZ-bm^kJ02L~3ACQnyCmvv4FO#meY6wUwu

literal 0
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diff --git a/docs/build/html/_static/pygments.css b/docs/build/html/_static/pygments.css
new file mode 100644
index 0000000..84ab303
--- /dev/null
+++ b/docs/build/html/_static/pygments.css
@@ -0,0 +1,75 @@
+pre { line-height: 125%; }
+td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+.highlight .hll { background-color: #ffffcc }
+.highlight { background: #f8f8f8; }
+.highlight .c { color: #3D7B7B; font-style: italic } /* Comment */
+.highlight .err { border: 1px solid #FF0000 } /* Error */
+.highlight .k { color: #008000; font-weight: bold } /* Keyword */
+.highlight .o { color: #666666 } /* Operator */
+.highlight .ch { color: #3D7B7B; font-style: italic } /* Comment.Hashbang */
+.highlight .cm { color: #3D7B7B; font-style: italic } /* Comment.Multiline */
+.highlight .cp { color: #9C6500 } /* Comment.Preproc */
+.highlight .cpf { color: #3D7B7B; font-style: italic } /* Comment.PreprocFile */
+.highlight .c1 { color: #3D7B7B; font-style: italic } /* Comment.Single */
+.highlight .cs { color: #3D7B7B; font-style: italic } /* Comment.Special */
+.highlight .gd { color: #A00000 } /* Generic.Deleted */
+.highlight .ge { font-style: italic } /* Generic.Emph */
+.highlight .ges { font-weight: bold; font-style: italic } /* Generic.EmphStrong */
+.highlight .gr { color: #E40000 } /* Generic.Error */
+.highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */
+.highlight .gi { color: #008400 } /* Generic.Inserted */
+.highlight .go { color: #717171 } /* Generic.Output */
+.highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */
+.highlight .gs { font-weight: bold } /* Generic.Strong */
+.highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */
+.highlight .gt { color: #0044DD } /* Generic.Traceback */
+.highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */
+.highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */
+.highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */
+.highlight .kp { color: #008000 } /* Keyword.Pseudo */
+.highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */
+.highlight .kt { color: #B00040 } /* Keyword.Type */
+.highlight .m { color: #666666 } /* Literal.Number */
+.highlight .s { color: #BA2121 } /* Literal.String */
+.highlight .na { color: #687822 } /* Name.Attribute */
+.highlight .nb { color: #008000 } /* Name.Builtin */
+.highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */
+.highlight .no { color: #880000 } /* Name.Constant */
+.highlight .nd { color: #AA22FF } /* Name.Decorator */
+.highlight .ni { color: #717171; font-weight: bold } /* Name.Entity */
+.highlight .ne { color: #CB3F38; font-weight: bold } /* Name.Exception */
+.highlight .nf { color: #0000FF } /* Name.Function */
+.highlight .nl { color: #767600 } /* Name.Label */
+.highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */
+.highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */
+.highlight .nv { color: #19177C } /* Name.Variable */
+.highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */
+.highlight .w { color: #bbbbbb } /* Text.Whitespace */
+.highlight .mb { color: #666666 } /* Literal.Number.Bin */
+.highlight .mf { color: #666666 } /* Literal.Number.Float */
+.highlight .mh { color: #666666 } /* Literal.Number.Hex */
+.highlight .mi { color: #666666 } /* Literal.Number.Integer */
+.highlight .mo { color: #666666 } /* Literal.Number.Oct */
+.highlight .sa { color: #BA2121 } /* Literal.String.Affix */
+.highlight .sb { color: #BA2121 } /* Literal.String.Backtick */
+.highlight .sc { color: #BA2121 } /* Literal.String.Char */
+.highlight .dl { color: #BA2121 } /* Literal.String.Delimiter */
+.highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */
+.highlight .s2 { color: #BA2121 } /* Literal.String.Double */
+.highlight .se { color: #AA5D1F; font-weight: bold } /* Literal.String.Escape */
+.highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */
+.highlight .si { color: #A45A77; font-weight: bold } /* Literal.String.Interpol */
+.highlight .sx { color: #008000 } /* Literal.String.Other */
+.highlight .sr { color: #A45A77 } /* Literal.String.Regex */
+.highlight .s1 { color: #BA2121 } /* Literal.String.Single */
+.highlight .ss { color: #19177C } /* Literal.String.Symbol */
+.highlight .bp { color: #008000 } /* Name.Builtin.Pseudo */
+.highlight .fm { color: #0000FF } /* Name.Function.Magic */
+.highlight .vc { color: #19177C } /* Name.Variable.Class */
+.highlight .vg { color: #19177C } /* Name.Variable.Global */
+.highlight .vi { color: #19177C } /* Name.Variable.Instance */
+.highlight .vm { color: #19177C } /* Name.Variable.Magic */
+.highlight .il { color: #666666 } /* Literal.Number.Integer.Long */
\ No newline at end of file
diff --git a/docs/build/html/_static/searchtools.js b/docs/build/html/_static/searchtools.js
new file mode 100644
index 0000000..b08d58c
--- /dev/null
+++ b/docs/build/html/_static/searchtools.js
@@ -0,0 +1,620 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2024 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+/**
+ * Simple result scoring code.
+ */
+if (typeof Scorer === "undefined") {
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [docname, title, anchor, descr, score, filename]
+    // and returns the new score.
+    /*
+    score: result => {
+      const [docname, title, anchor, descr, score, filename] = result
+      return score
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {
+      0: 15, // used to be importantResults
+      1: 5, // used to be objectResults
+      2: -5, // used to be unimportantResults
+    },
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2,
+  };
+}
+
+const _removeChildren = (element) => {
+  while (element && element.lastChild) element.removeChild(element.lastChild);
+};
+
+/**
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
+ */
+const _escapeRegExp = (string) =>
+  string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
+
+const _displayItem = (item, searchTerms, highlightTerms) => {
+  const docBuilder = DOCUMENTATION_OPTIONS.BUILDER;
+  const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+  const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+  const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY;
+  const contentRoot = document.documentElement.dataset.content_root;
+
+  const [docName, title, anchor, descr, score, _filename] = item;
+
+  let listItem = document.createElement("li");
+  let requestUrl;
+  let linkUrl;
+  if (docBuilder === "dirhtml") {
+    // dirhtml builder
+    let dirname = docName + "/";
+    if (dirname.match(/\/index\/$/))
+      dirname = dirname.substring(0, dirname.length - 6);
+    else if (dirname === "index/") dirname = "";
+    requestUrl = contentRoot + dirname;
+    linkUrl = requestUrl;
+  } else {
+    // normal html builders
+    requestUrl = contentRoot + docName + docFileSuffix;
+    linkUrl = docName + docLinkSuffix;
+  }
+  let linkEl = listItem.appendChild(document.createElement("a"));
+  linkEl.href = linkUrl + anchor;
+  linkEl.dataset.score = score;
+  linkEl.innerHTML = title;
+  if (descr) {
+    listItem.appendChild(document.createElement("span")).innerHTML =
+      " (" + descr + ")";
+    // highlight search terms in the description
+    if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+      highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+  }
+  else if (showSearchSummary)
+    fetch(requestUrl)
+      .then((responseData) => responseData.text())
+      .then((data) => {
+        if (data)
+          listItem.appendChild(
+            Search.makeSearchSummary(data, searchTerms, anchor)
+          );
+        // highlight search terms in the summary
+        if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+          highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+      });
+  Search.output.appendChild(listItem);
+};
+const _finishSearch = (resultCount) => {
+  Search.stopPulse();
+  Search.title.innerText = _("Search Results");
+  if (!resultCount)
+    Search.status.innerText = Documentation.gettext(
+      "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories."
+    );
+  else
+    Search.status.innerText = _(
+      "Search finished, found ${resultCount} page(s) matching the search query."
+    ).replace('${resultCount}', resultCount);
+};
+const _displayNextItem = (
+  results,
+  resultCount,
+  searchTerms,
+  highlightTerms,
+) => {
+  // results left, load the summary and display it
+  // this is intended to be dynamic (don't sub resultsCount)
+  if (results.length) {
+    _displayItem(results.pop(), searchTerms, highlightTerms);
+    setTimeout(
+      () => _displayNextItem(results, resultCount, searchTerms, highlightTerms),
+      5
+    );
+  }
+  // search finished, update title and status message
+  else _finishSearch(resultCount);
+};
+// Helper function used by query() to order search results.
+// Each input is an array of [docname, title, anchor, descr, score, filename].
+// Order the results by score (in opposite order of appearance, since the
+// `_displayNextItem` function uses pop() to retrieve items) and then alphabetically.
+const _orderResultsByScoreThenName = (a, b) => {
+  const leftScore = a[4];
+  const rightScore = b[4];
+  if (leftScore === rightScore) {
+    // same score: sort alphabetically
+    const leftTitle = a[1].toLowerCase();
+    const rightTitle = b[1].toLowerCase();
+    if (leftTitle === rightTitle) return 0;
+    return leftTitle > rightTitle ? -1 : 1; // inverted is intentional
+  }
+  return leftScore > rightScore ? 1 : -1;
+};
+
+/**
+ * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
+ * custom function per language.
+ *
+ * The regular expression works by splitting the string on consecutive characters
+ * that are not Unicode letters, numbers, underscores, or emoji characters.
+ * This is the same as ``\W+`` in Python, preserving the surrogate pair area.
+ */
+if (typeof splitQuery === "undefined") {
+  var splitQuery = (query) => query
+      .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu)
+      .filter(term => term)  // remove remaining empty strings
+}
+
+/**
+ * Search Module
+ */
+const Search = {
+  _index: null,
+  _queued_query: null,
+  _pulse_status: -1,
+
+  htmlToText: (htmlString, anchor) => {
+    const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html');
+    for (const removalQuery of [".headerlink", "script", "style"]) {
+      htmlElement.querySelectorAll(removalQuery).forEach((el) => { el.remove() });
+    }
+    if (anchor) {
+      const anchorContent = htmlElement.querySelector(`[role="main"] ${anchor}`);
+      if (anchorContent) return anchorContent.textContent;
+
+      console.warn(
+        `Anchored content block not found. Sphinx search tries to obtain it via DOM query '[role=main] ${anchor}'. Check your theme or template.`
+      );
+    }
+
+    // if anchor not specified or not found, fall back to main content
+    const docContent = htmlElement.querySelector('[role="main"]');
+    if (docContent) return docContent.textContent;
+
+    console.warn(
+      "Content block not found. Sphinx search tries to obtain it via DOM query '[role=main]'. Check your theme or template."
+    );
+    return "";
+  },
+
+  init: () => {
+    const query = new URLSearchParams(window.location.search).get("q");
+    document
+      .querySelectorAll('input[name="q"]')
+      .forEach((el) => (el.value = query));
+    if (query) Search.performSearch(query);
+  },
+
+  loadIndex: (url) =>
+    (document.body.appendChild(document.createElement("script")).src = url),
+
+  setIndex: (index) => {
+    Search._index = index;
+    if (Search._queued_query !== null) {
+      const query = Search._queued_query;
+      Search._queued_query = null;
+      Search.query(query);
+    }
+  },
+
+  hasIndex: () => Search._index !== null,
+
+  deferQuery: (query) => (Search._queued_query = query),
+
+  stopPulse: () => (Search._pulse_status = -1),
+
+  startPulse: () => {
+    if (Search._pulse_status >= 0) return;
+
+    const pulse = () => {
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      Search.dots.innerText = ".".repeat(Search._pulse_status);
+      if (Search._pulse_status >= 0) window.setTimeout(pulse, 500);
+    };
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch: (query) => {
+    // create the required interface elements
+    const searchText = document.createElement("h2");
+    searchText.textContent = _("Searching");
+    const searchSummary = document.createElement("p");
+    searchSummary.classList.add("search-summary");
+    searchSummary.innerText = "";
+    const searchList = document.createElement("ul");
+    searchList.classList.add("search");
+
+    const out = document.getElementById("search-results");
+    Search.title = out.appendChild(searchText);
+    Search.dots = Search.title.appendChild(document.createElement("span"));
+    Search.status = out.appendChild(searchSummary);
+    Search.output = out.appendChild(searchList);
+
+    const searchProgress = document.getElementById("search-progress");
+    // Some themes don't use the search progress node
+    if (searchProgress) {
+      searchProgress.innerText = _("Preparing search...");
+    }
+    Search.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (Search.hasIndex()) Search.query(query);
+    else Search.deferQuery(query);
+  },
+
+  _parseQuery: (query) => {
+    // stem the search terms and add them to the correct list
+    const stemmer = new Stemmer();
+    const searchTerms = new Set();
+    const excludedTerms = new Set();
+    const highlightTerms = new Set();
+    const objectTerms = new Set(splitQuery(query.toLowerCase().trim()));
+    splitQuery(query.trim()).forEach((queryTerm) => {
+      const queryTermLower = queryTerm.toLowerCase();
+
+      // maybe skip this "word"
+      // stopwords array is from language_data.js
+      if (
+        stopwords.indexOf(queryTermLower) !== -1 ||
+        queryTerm.match(/^\d+$/)
+      )
+        return;
+
+      // stem the word
+      let word = stemmer.stemWord(queryTermLower);
+      // select the correct list
+      if (word[0] === "-") excludedTerms.add(word.substr(1));
+      else {
+        searchTerms.add(word);
+        highlightTerms.add(queryTermLower);
+      }
+    });
+
+    if (SPHINX_HIGHLIGHT_ENABLED) {  // set in sphinx_highlight.js
+      localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" "))
+    }
+
+    // console.debug("SEARCH: searching for:");
+    // console.info("required: ", [...searchTerms]);
+    // console.info("excluded: ", [...excludedTerms]);
+
+    return [query, searchTerms, excludedTerms, highlightTerms, objectTerms];
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  _performSearch: (query, searchTerms, excludedTerms, highlightTerms, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+    const allTitles = Search._index.alltitles;
+    const indexEntries = Search._index.indexentries;
+
+    // Collect multiple result groups to be sorted separately and then ordered.
+    // Each is an array of [docname, title, anchor, descr, score, filename].
+    const normalResults = [];
+    const nonMainIndexResults = [];
+
+    _removeChildren(document.getElementById("search-progress"));
+
+    const queryLower = query.toLowerCase().trim();
+    for (const [title, foundTitles] of Object.entries(allTitles)) {
+      if (title.toLowerCase().trim().includes(queryLower) && (queryLower.length >= title.length/2)) {
+        for (const [file, id] of foundTitles) {
+          const score = Math.round(Scorer.title * queryLower.length / title.length);
+          const boost = titles[file] === title ? 1 : 0;  // add a boost for document titles
+          normalResults.push([
+            docNames[file],
+            titles[file] !== title ? `${titles[file]} > ${title}` : title,
+            id !== null ? "#" + id : "",
+            null,
+            score + boost,
+            filenames[file],
+          ]);
+        }
+      }
+    }
+
+    // search for explicit entries in index directives
+    for (const [entry, foundEntries] of Object.entries(indexEntries)) {
+      if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) {
+        for (const [file, id, isMain] of foundEntries) {
+          const score = Math.round(100 * queryLower.length / entry.length);
+          const result = [
+            docNames[file],
+            titles[file],
+            id ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ];
+          if (isMain) {
+            normalResults.push(result);
+          } else {
+            nonMainIndexResults.push(result);
+          }
+        }
+      }
+    }
+
+    // lookup as object
+    objectTerms.forEach((term) =>
+      normalResults.push(...Search.performObjectSearch(term, objectTerms))
+    );
+
+    // lookup as search terms in fulltext
+    normalResults.push(...Search.performTermsSearch(searchTerms, excludedTerms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) {
+      normalResults.forEach((item) => (item[4] = Scorer.score(item)));
+      nonMainIndexResults.forEach((item) => (item[4] = Scorer.score(item)));
+    }
+
+    // Sort each group of results by score and then alphabetically by name.
+    normalResults.sort(_orderResultsByScoreThenName);
+    nonMainIndexResults.sort(_orderResultsByScoreThenName);
+
+    // Combine the result groups in (reverse) order.
+    // Non-main index entries are typically arbitrary cross-references,
+    // so display them after other results.
+    let results = [...nonMainIndexResults, ...normalResults];
+
+    // remove duplicate search results
+    // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept
+    let seen = new Set();
+    results = results.reverse().reduce((acc, result) => {
+      let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(',');
+      if (!seen.has(resultStr)) {
+        acc.push(result);
+        seen.add(resultStr);
+      }
+      return acc;
+    }, []);
+
+    return results.reverse();
+  },
+
+  query: (query) => {
+    const [searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms] = Search._parseQuery(query);
+    const results = Search._performSearch(searchQuery, searchTerms, excludedTerms, highlightTerms, objectTerms);
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    // console.info("search results:", Search.lastresults);
+
+    // print the results
+    _displayNextItem(results, results.length, searchTerms, highlightTerms);
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch: (object, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const objects = Search._index.objects;
+    const objNames = Search._index.objnames;
+    const titles = Search._index.titles;
+
+    const results = [];
+
+    const objectSearchCallback = (prefix, match) => {
+      const name = match[4]
+      const fullname = (prefix ? prefix + "." : "") + name;
+      const fullnameLower = fullname.toLowerCase();
+      if (fullnameLower.indexOf(object) < 0) return;
+
+      let score = 0;
+      const parts = fullnameLower.split(".");
+
+      // check for different match types: exact matches of full name or
+      // "last name" (i.e. last dotted part)
+      if (fullnameLower === object || parts.slice(-1)[0] === object)
+        score += Scorer.objNameMatch;
+      else if (parts.slice(-1)[0].indexOf(object) > -1)
+        score += Scorer.objPartialMatch; // matches in last name
+
+      const objName = objNames[match[1]][2];
+      const title = titles[match[0]];
+
+      // If more than one term searched for, we require other words to be
+      // found in the name/title/description
+      const otherTerms = new Set(objectTerms);
+      otherTerms.delete(object);
+      if (otherTerms.size > 0) {
+        const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase();
+        if (
+          [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0)
+        )
+          return;
+      }
+
+      let anchor = match[3];
+      if (anchor === "") anchor = fullname;
+      else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname;
+
+      const descr = objName + _(", in ") + title;
+
+      // add custom score for some objects according to scorer
+      if (Scorer.objPrio.hasOwnProperty(match[2]))
+        score += Scorer.objPrio[match[2]];
+      else score += Scorer.objPrioDefault;
+
+      results.push([
+        docNames[match[0]],
+        fullname,
+        "#" + anchor,
+        descr,
+        score,
+        filenames[match[0]],
+      ]);
+    };
+    Object.keys(objects).forEach((prefix) =>
+      objects[prefix].forEach((array) =>
+        objectSearchCallback(prefix, array)
+      )
+    );
+    return results;
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch: (searchTerms, excludedTerms) => {
+    // prepare search
+    const terms = Search._index.terms;
+    const titleTerms = Search._index.titleterms;
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+
+    const scoreMap = new Map();
+    const fileMap = new Map();
+
+    // perform the search on the required terms
+    searchTerms.forEach((word) => {
+      const files = [];
+      const arr = [
+        { files: terms[word], score: Scorer.term },
+        { files: titleTerms[word], score: Scorer.title },
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        const escapedWord = _escapeRegExp(word);
+        if (!terms.hasOwnProperty(word)) {
+          Object.keys(terms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: terms[term], score: Scorer.partialTerm });
+          });
+        }
+        if (!titleTerms.hasOwnProperty(word)) {
+          Object.keys(titleTerms).forEach((term) => {
+            if (term.match(escapedWord))
+              arr.push({ files: titleTerms[term], score: Scorer.partialTitle });
+          });
+        }
+      }
+
+      // no match but word was a required one
+      if (arr.every((record) => record.files === undefined)) return;
+
+      // found search word in contents
+      arr.forEach((record) => {
+        if (record.files === undefined) return;
+
+        let recordFiles = record.files;
+        if (recordFiles.length === undefined) recordFiles = [recordFiles];
+        files.push(...recordFiles);
+
+        // set score for the word in each file
+        recordFiles.forEach((file) => {
+          if (!scoreMap.has(file)) scoreMap.set(file, {});
+          scoreMap.get(file)[word] = record.score;
+        });
+      });
+
+      // create the mapping
+      files.forEach((file) => {
+        if (!fileMap.has(file)) fileMap.set(file, [word]);
+        else if (fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word);
+      });
+    });
+
+    // now check if the files don't contain excluded terms
+    const results = [];
+    for (const [file, wordList] of fileMap) {
+      // check if all requirements are matched
+
+      // as search terms with length < 3 are discarded
+      const filteredTermCount = [...searchTerms].filter(
+        (term) => term.length > 2
+      ).length;
+      if (
+        wordList.length !== searchTerms.size &&
+        wordList.length !== filteredTermCount
+      )
+        continue;
+
+      // ensure that none of the excluded terms is in the search result
+      if (
+        [...excludedTerms].some(
+          (term) =>
+            terms[term] === file ||
+            titleTerms[term] === file ||
+            (terms[term] || []).includes(file) ||
+            (titleTerms[term] || []).includes(file)
+        )
+      )
+        break;
+
+      // select one (max) score for the file.
+      const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w]));
+      // add result to the result list
+      results.push([
+        docNames[file],
+        titles[file],
+        "",
+        null,
+        score,
+        filenames[file],
+      ]);
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words.
+   */
+  makeSearchSummary: (htmlText, keywords, anchor) => {
+    const text = Search.htmlToText(htmlText, anchor);
+    if (text === "") return null;
+
+    const textLower = text.toLowerCase();
+    const actualStartPosition = [...keywords]
+      .map((k) => textLower.indexOf(k.toLowerCase()))
+      .filter((i) => i > -1)
+      .slice(-1)[0];
+    const startWithContext = Math.max(actualStartPosition - 120, 0);
+
+    const top = startWithContext === 0 ? "" : "...";
+    const tail = startWithContext + 240 < text.length ? "..." : "";
+
+    let summary = document.createElement("p");
+    summary.classList.add("context");
+    summary.textContent = top + text.substr(startWithContext, 240).trim() + tail;
+
+    return summary;
+  },
+};
+
+_ready(Search.init);
diff --git a/docs/build/html/_static/sphinx_highlight.js b/docs/build/html/_static/sphinx_highlight.js
new file mode 100644
index 0000000..8a96c69
--- /dev/null
+++ b/docs/build/html/_static/sphinx_highlight.js
@@ -0,0 +1,154 @@
+/* Highlighting utilities for Sphinx HTML documentation. */
+"use strict";
+
+const SPHINX_HIGHLIGHT_ENABLED = true
+
+/**
+ * highlight a given string on a node by wrapping it in
+ * span elements with the given class name.
+ */
+const _highlight = (node, addItems, text, className) => {
+  if (node.nodeType === Node.TEXT_NODE) {
+    const val = node.nodeValue;
+    const parent = node.parentNode;
+    const pos = val.toLowerCase().indexOf(text);
+    if (
+      pos >= 0 &&
+      !parent.classList.contains(className) &&
+      !parent.classList.contains("nohighlight")
+    ) {
+      let span;
+
+      const closestNode = parent.closest("body, svg, foreignObject");
+      const isInSVG = closestNode && closestNode.matches("svg");
+      if (isInSVG) {
+        span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+      } else {
+        span = document.createElement("span");
+        span.classList.add(className);
+      }
+
+      span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+      const rest = document.createTextNode(val.substr(pos + text.length));
+      parent.insertBefore(
+        span,
+        parent.insertBefore(
+          rest,
+          node.nextSibling
+        )
+      );
+      node.nodeValue = val.substr(0, pos);
+      /* There may be more occurrences of search term in this node. So call this
+       * function recursively on the remaining fragment.
+       */
+      _highlight(rest, addItems, text, className);
+
+      if (isInSVG) {
+        const rect = document.createElementNS(
+          "http://www.w3.org/2000/svg",
+          "rect"
+        );
+        const bbox = parent.getBBox();
+        rect.x.baseVal.value = bbox.x;
+        rect.y.baseVal.value = bbox.y;
+        rect.width.baseVal.value = bbox.width;
+        rect.height.baseVal.value = bbox.height;
+        rect.setAttribute("class", className);
+        addItems.push({ parent: parent, target: rect });
+      }
+    }
+  } else if (node.matches && !node.matches("button, select, textarea")) {
+    node.childNodes.forEach((el) => _highlight(el, addItems, text, className));
+  }
+};
+const _highlightText = (thisNode, text, className) => {
+  let addItems = [];
+  _highlight(thisNode, addItems, text, className);
+  addItems.forEach((obj) =>
+    obj.parent.insertAdjacentElement("beforebegin", obj.target)
+  );
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const SphinxHighlight = {
+
+  /**
+   * highlight the search words provided in localstorage in the text
+   */
+  highlightSearchWords: () => {
+    if (!SPHINX_HIGHLIGHT_ENABLED) return;  // bail if no highlight
+
+    // get and clear terms from localstorage
+    const url = new URL(window.location);
+    const highlight =
+        localStorage.getItem("sphinx_highlight_terms")
+        || url.searchParams.get("highlight")
+        || "";
+    localStorage.removeItem("sphinx_highlight_terms")
+    url.searchParams.delete("highlight");
+    window.history.replaceState({}, "", url);
+
+    // get individual terms from highlight string
+    const terms = highlight.toLowerCase().split(/\s+/).filter(x => x);
+    if (terms.length === 0) return; // nothing to do
+
+    // There should never be more than one element matching "div.body"
+    const divBody = document.querySelectorAll("div.body");
+    const body = divBody.length ? divBody[0] : document.querySelector("body");
+    window.setTimeout(() => {
+      terms.forEach((term) => _highlightText(body, term, "highlighted"));
+    }, 10);
+
+    const searchBox = document.getElementById("searchbox");
+    if (searchBox === null) return;
+    searchBox.appendChild(
+      document
+        .createRange()
+        .createContextualFragment(
+          '<p class="highlight-link">' +
+            '<a href="javascript:SphinxHighlight.hideSearchWords()">' +
+            _("Hide Search Matches") +
+            "</a></p>"
+        )
+    );
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords: () => {
+    document
+      .querySelectorAll("#searchbox .highlight-link")
+      .forEach((el) => el.remove());
+    document
+      .querySelectorAll("span.highlighted")
+      .forEach((el) => el.classList.remove("highlighted"));
+    localStorage.removeItem("sphinx_highlight_terms")
+  },
+
+  initEscapeListener: () => {
+    // only install a listener if it is really needed
+    if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return;
+      if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) {
+        SphinxHighlight.hideSearchWords();
+        event.preventDefault();
+      }
+    });
+  },
+};
+
+_ready(() => {
+  /* Do not call highlightSearchWords() when we are on the search page.
+   * It will highlight words from the *previous* search query.
+   */
+  if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords();
+  SphinxHighlight.initEscapeListener();
+});
diff --git a/docs/build/html/example.html b/docs/build/html/example.html
new file mode 100644
index 0000000..9480cc6
--- /dev/null
+++ b/docs/build/html/example.html
@@ -0,0 +1,220 @@
+<!DOCTYPE html>
+<html class="writer-html5" lang="en" data-content_root="./">
+<head>
+  <meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+  <meta name="viewport" content="width=device-width, initial-scale=1.0" />
+  <title>Example &mdash; Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model September, 2024 documentation</title>
+      <link rel="stylesheet" type="text/css" href="_static/pygments.css?v=80d5e7a1" />
+      <link rel="stylesheet" type="text/css" href="_static/css/theme.css?v=19f00094" />
+      <link rel="stylesheet" type="text/css" href="_static/copybutton.css?v=76b2166b" />
+
+  
+  <!--[if lt IE 9]>
+    <script src="_static/js/html5shiv.min.js"></script>
+  <![endif]-->
+  
+        <script src="_static/jquery.js?v=5d32c60e"></script>
+        <script src="_static/_sphinx_javascript_frameworks_compat.js?v=2cd50e6c"></script>
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+    <nav data-toggle="wy-nav-shift" class="wy-nav-side">
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+          <a href="index.html" class="icon icon-home">
+            Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model
+          </a>
+<div role="search">
+  <form id="rtd-search-form" class="wy-form" action="search.html" method="get">
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+              <p class="caption" role="heading"><span class="caption-text">Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:</span></p>
+<ul class="current">
+<li class="toctree-l1"><a class="reference internal" href="src.html">Source Code</a></li>
+<li class="toctree-l1 current"><a class="current reference internal" href="#">Example</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="#import-fenicsx-implementation-and-necessary-libraries">Import FEniCSx implementation and necessary libraries</a></li>
+<li class="toctree-l2"><a class="reference internal" href="#define-parameters-and-boundary-conditions">Define parameters and boundary conditions</a></li>
+<li class="toctree-l2"><a class="reference internal" href="#define-mesh-and-solver-settings">Define mesh and solver settings</a></li>
+<li class="toctree-l2"><a class="reference internal" href="#solve-the-system">Solve the system</a></li>
+<li class="toctree-l2"><a class="reference internal" href="#visualize-the-results">Visualize the results</a><ul>
+<li class="toctree-l3"><a class="reference internal" href="#the-electric-potential">The electric potential</a></li>
+<li class="toctree-l3"><a class="reference internal" href="#the-pressure">The pressure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="#the-atomic-fractions">The atomic fractions</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+
+        </div>
+      </div>
+    </nav>
+
+    <section data-toggle="wy-nav-shift" class="wy-nav-content-wrap"><nav class="wy-nav-top" aria-label="Mobile navigation menu" >
+          <i data-toggle="wy-nav-top" class="fa fa-bars"></i>
+          <a href="index.html">Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model</a>
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+
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+      <li class="breadcrumb-item active">Example</li>
+      <li class="wy-breadcrumbs-aside">
+            <a href="_sources/example.rst.txt" rel="nofollow"> View page source</a>
+      </li>
+  </ul>
+  <hr/>
+</div>
+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+           <div itemprop="articleBody">
+             
+  <section id="example">
+<h1>Example<a class="headerlink" href="#example" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">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”</div>
+</div>
+<div class="line-block">
+<div class="line">The code will work if executed in the examples/ReproducableCode folder, as the TernaryElectrolyte.py file is located in the same folder</div>
+</div>
+<section id="import-fenicsx-implementation-and-necessary-libraries">
+<h2>Import FEniCSx implementation and necessary libraries<a class="headerlink" href="#import-fenicsx-implementation-and-necessary-libraries" title="Link to this heading"></a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">sys</span>
+<span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">src_path</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="vm">__file__</span><span class="p">)),</span> <span class="s1">&#39;../..&#39;</span><span class="p">,</span> <span class="s1">&#39;src&#39;</span><span class="p">)</span>
+<span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">src_path</span><span class="p">)</span>
+
+<span class="kn">from</span> <span class="nn">Eq04</span> <span class="kn">import</span> <span class="n">solve_System_4eq</span>
+
+<span class="k">del</span> <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+</pre></div>
+</div>
+</section>
+<section id="define-parameters-and-boundary-conditions">
+<h2>Define parameters and boundary conditions<a class="headerlink" href="#define-parameters-and-boundary-conditions" title="Link to this heading"></a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">phi_left</span> <span class="o">=</span> <span class="mf">8.0</span>
+<span class="n">phi_right</span> <span class="o">=</span> <span class="mf">0.0</span>
+<span class="n">p_right</span> <span class="o">=</span> <span class="mf">0.0</span>
+<span class="n">y_A_R</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span>
+<span class="n">y_C_R</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span>
+<span class="n">z_A</span> <span class="o">=</span> <span class="o">-</span><span class="mf">1.0</span>
+<span class="n">z_C</span> <span class="o">=</span> <span class="mf">1.0</span>
+<span class="n">K</span> <span class="o">=</span> <span class="s1">&#39;incompressible&#39;</span>
+<span class="n">Lambda2</span> <span class="o">=</span> <span class="mf">8.553e-6</span>
+<span class="n">a2</span> <span class="o">=</span> <span class="mf">7.5412e-4</span>
+</pre></div>
+</div>
+</section>
+<section id="define-mesh-and-solver-settings">
+<h2>Define mesh and solver settings<a class="headerlink" href="#define-mesh-and-solver-settings" title="Link to this heading"></a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">number_cells</span> <span class="o">=</span> <span class="mi">1024</span>
+<span class="n">refinement_style</span> <span class="o">=</span> <span class="s1">&#39;log&#39;</span>
+<span class="n">rtol</span> <span class="o">=</span> <span class="mf">1e-8</span>
+</pre></div>
+</div>
+</section>
+<section id="solve-the-system">
+<h2>Solve the system<a class="headerlink" href="#solve-the-system" title="Link to this heading"></a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">y_A</span><span class="p">,</span> <span class="n">y_C</span><span class="p">,</span> <span class="n">phi</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">solve_System_4eq</span><span class="p">(</span><span class="n">phi_left</span><span class="p">,</span> <span class="n">phi_right</span><span class="p">,</span> <span class="n">p_right</span><span class="p">,</span> <span class="n">z_A</span><span class="p">,</span> <span class="n">z_C</span><span class="p">,</span> <span class="n">y_A_R</span><span class="p">,</span> <span class="n">y_C_R</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">Lambda2</span><span class="p">,</span> <span class="n">a2</span><span class="p">,</span> <span class="n">number_cells</span><span class="p">,</span> <span class="n">relax_param</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">x1</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>    <span class="n">refinement_style</span><span class="o">=</span><span class="s1">&#39;hard_log&#39;</span><span class="p">,</span> <span class="n">return_type</span><span class="o">=</span><span class="s1">&#39;Vector&#39;</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">1_000</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="n">rtol</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="visualize-the-results">
+<h2>Visualize the results<a class="headerlink" href="#visualize-the-results" title="Link to this heading"></a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$</span><span class="se">\\</span><span class="s1">varphi$  [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.05</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$p$ [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.05</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y_A</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$y_A$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y_C</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$y_C$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y_A</span> <span class="o">-</span> <span class="n">y_C</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$y_S$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;$y_</span><span class="se">\\</span><span class="s1">alpha$ [-]&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.05</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+<section id="the-electric-potential">
+<h3>The electric potential<a class="headerlink" href="#the-electric-potential" title="Link to this heading"></a></h3>
+<a class="reference internal image-reference" href="_images/TernaryPotential.svg"><img alt="_images/TernaryPotential.svg" src="_images/TernaryPotential.svg" style="width: 600px;" /></a>
+</section>
+<section id="the-pressure">
+<h3>The pressure<a class="headerlink" href="#the-pressure" title="Link to this heading"></a></h3>
+<a class="reference internal image-reference" href="_images/TernaryPressure.svg"><img alt="_images/TernaryPressure.svg" src="_images/TernaryPressure.svg" style="width: 600px;" /></a>
+</section>
+<section id="the-atomic-fractions">
+<h3>The atomic fractions<a class="headerlink" href="#the-atomic-fractions" title="Link to this heading"></a></h3>
+<a class="reference internal image-reference" href="_images/TernaryConcentrations.svg"><img alt="_images/TernaryConcentrations.svg" src="_images/TernaryConcentrations.svg" style="width: 600px;" /></a>
+</section>
+</section>
+</section>
+
+
+           </div>
+          </div>
+          <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+        <a href="src.html" class="btn btn-neutral float-left" title="Source Code" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a>
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+    <p>&#169; Copyright 2024, Jan Habscheid.</p>
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+  <title>Index &mdash; Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model September, 2024 documentation</title>
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+    <nav data-toggle="wy-nav-shift" class="wy-nav-side">
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+          
+          
+          <a href="index.html" class="icon icon-home">
+            Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model
+          </a>
+<div role="search">
+  <form id="rtd-search-form" class="wy-form" action="search.html" method="get">
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+              <p class="caption" role="heading"><span class="caption-text">Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="src.html">Source Code</a></li>
+<li class="toctree-l1"><a class="reference internal" href="example.html">Example</a></li>
+</ul>
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+          <a href="index.html">Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model</a>
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+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+           <div itemprop="articleBody">
+             
+
+<h1 id="index">Index</h1>
+
+<div class="genindex-jumpbox">
+ <a href="#C"><strong>C</strong></a>
+ | <a href="#D"><strong>D</strong></a>
+ | <a href="#E"><strong>E</strong></a>
+ | <a href="#H"><strong>H</strong></a>
+ | <a href="#M"><strong>M</strong></a>
+ | <a href="#N"><strong>N</strong></a>
+ | <a href="#P"><strong>P</strong></a>
+ | <a href="#Q"><strong>Q</strong></a>
+ | <a href="#S"><strong>S</strong></a>
+ 
+</div>
+<h2 id="C">C</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.C_dl">C_dl() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Eq04.create_refined_mesh">create_refined_mesh() (in module Eq04)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="D">D</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.dx">dx() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="E">E</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#ElectrolyticDiode.ElectrolyticDiode">ElectrolyticDiode() (in module ElectrolyticDiode)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="H">H</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    Helper_DoubleLayerCapacity
+
+      <ul>
+        <li><a href="src.html#module-Helper_DoubleLayerCapacity">module</a>
+</li>
+      </ul></li>
+  </ul></td>
+</tr></table>
+
+<h2 id="M">M</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    module
+
+      <ul>
+        <li><a href="src.html#module-Helper_DoubleLayerCapacity">Helper_DoubleLayerCapacity</a>
+</li>
+      </ul></li>
+  </ul></td>
+</tr></table>
+
+<h2 id="N">N</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.n">n() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="P">P</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.Phi_pot_center">Phi_pot_center() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="Q">Q</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.Q_DL_dim_ana">Q_DL_dim_ana() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.Q_DL_dimless_ana">Q_DL_dimless_ana() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.Q_num_">Q_num_() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+      <li><a href="src.html#Helper_DoubleLayerCapacity.Q_num_dim">Q_num_dim() (in module Helper_DoubleLayerCapacity)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+<h2 id="S">S</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Eq02.solve_System_2eq">solve_System_2eq() (in module Eq02)</a>
+</li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li><a href="src.html#Eq04.solve_System_4eq">solve_System_4eq() (in module Eq04)</a>
+</li>
+      <li><a href="src.html#EqN.solve_System_Neq">solve_System_Neq() (in module EqN)</a>
+</li>
+  </ul></td>
+</tr></table>
+
+
+
+           </div>
+          </div>
+          <footer>
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+<body class="wy-body-for-nav"> 
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+              <p class="caption" role="heading"><span class="caption-text">Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="src.html">Source Code</a></li>
+<li class="toctree-l1"><a class="reference internal" href="example.html">Example</a></li>
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+      <li class="breadcrumb-item active">Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation</li>
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+            <a href="_sources/index.rst.txt" rel="nofollow"> View page source</a>
+      </li>
+  </ul>
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+          <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
+           <div itemprop="articleBody">
+             
+  <section id="reproducibility-repository-for-numerical-treatment-of-a-thermodynamically-consistent-electrolyte-model-documentation">
+<h1>Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation<a class="headerlink" href="#reproducibility-repository-for-numerical-treatment-of-a-thermodynamically-consistent-electrolyte-model-documentation" title="Link to this heading"></a></h1>
+<div class="admonition- admonition">
+<p class="admonition-title"></p>
+<p>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: ToDo Add DOI</p>
+</div>
+</section>
+<section id="thesis">
+<h1>Thesis<a class="headerlink" href="#thesis" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">This repository contains the code to reproduce the results presented in the bachelor thesis: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model</div>
+<div class="line">Find the thesis at:</div>
+</div>
+<ul class="simple">
+<li><p>Insert link to thesis</p></li>
+</ul>
+</section>
+<section id="abstract">
+<h1>Abstract<a class="headerlink" href="#abstract" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">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.</div>
+</div>
+<div class="line-block">
+<div class="line">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.</div>
+</div>
+<div class="line-block">
+<div class="line">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.</div>
+</div>
+</section>
+<section id="installation">
+<h1>Installation<a class="headerlink" href="#installation" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">As a numerical solver, mainly FEniCSx was used and installed via conda.</div>
+<div class="line">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.</div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">create</span> <span class="o">--</span><span class="n">name</span> <span class="n">fenicsx</span><span class="o">-</span><span class="n">env</span> <span class="n">python</span><span class="o">=</span><span class="mf">3.12.3</span> <span class="o">-</span><span class="n">y</span>
+<span class="n">conda</span> <span class="n">activate</span> <span class="n">fenicsx</span><span class="o">-</span><span class="n">env</span>
+<span class="n">conda</span> <span class="n">install</span> <span class="o">-</span><span class="n">c</span> <span class="n">conda</span><span class="o">-</span><span class="n">forge</span> <span class="n">fenics</span><span class="o">-</span><span class="n">dolfinx</span><span class="o">=</span><span class="mf">0.8.0</span> <span class="n">mpich</span><span class="o">=</span><span class="mf">4.2.1</span> <span class="n">pyvista</span><span class="o">=</span><span class="mf">0.43.10</span> <span class="n">matplotlib</span><span class="o">=</span><span class="mf">3.8.4</span> <span class="n">numpy</span><span class="o">=</span><span class="mf">1.26.4</span> <span class="n">scipy</span><span class="o">=</span><span class="mf">1.14.0</span> <span class="o">-</span><span class="n">y</span>
+</pre></div>
+</div>
+</section>
+<section id="alternative-installation">
+<h1>Alternative installation<a class="headerlink" href="#alternative-installation" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">Use the “environment.yml” file to install all necessary environments</div>
+</div>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">conda</span> <span class="n">env</span> <span class="n">create</span> <span class="o">-</span><span class="n">f</span> <span class="n">environment</span><span class="o">.</span><span class="n">yml</span>
+</pre></div>
+</div>
+</section>
+<section id="usage">
+<h1>Usage<a class="headerlink" href="#usage" title="Link to this heading"></a></h1>
+<div class="line-block">
+<div class="line">Find the visualizations from the thesis and some extra calculations in the “examples” folder.</div>
+<div class="line">In the subfolder “ReproducableCode” is the code, to execute the calculations with some first visualizations.</div>
+<div class="line">The subfolder “Data” stores the data for all the simulations in a “.npz” file, which can be read with numpy <cite>np.load(file.npz)</cite>. “Visualizations” creates the necessary figures from the thesis and stores them in “.svg” format in “Figures”.</div>
+<div class="line">In “src” there are the generic FEniCSx implementations, that were used to calculate the examples.</div>
+</div>
+</section>
+<section id="contact">
+<h1>Contact<a class="headerlink" href="#contact" title="Link to this heading"></a></h1>
+<p><strong>Author</strong></p>
+<ul class="simple">
+<li><p>Jan Habscheid</p></li>
+<li><p><a class="reference external" href="mailto:Jan&#46;Habscheid&#37;&#52;&#48;rwth-aachen&#46;de">Jan<span>&#46;</span>Habscheid<span>&#64;</span>rwth-aachen<span>&#46;</span>de</a></p></li>
+</ul>
+<p><strong>Supervisor</strong></p>
+<ul class="simple">
+<li><p>Dr. Lambert Theissen</p></li>
+<li><p>ACoM - Applied and Computational Mathematics</p></li>
+<li><p>RWTH Aachen University</p></li>
+<li><p><a class="reference external" href="mailto:theisen&#37;&#52;&#48;acom&#46;rwth-aachen&#46;de">theisen<span>&#64;</span>acom<span>&#46;</span>rwth-aachen<span>&#46;</span>de</a></p></li>
+</ul>
+<p><strong>Supervisor</strong></p>
+<ul class="simple">
+<li><p>Prof. Dr. Manuel Torrilhon</p></li>
+<li><p>ACoM - Applied and Computational Mathematics</p></li>
+<li><p>RWTH Aachen University</p></li>
+<li><p><a class="reference external" href="mailto:mt&#37;&#52;&#48;acom&#46;rwth-aachen&#46;de">mt<span>&#64;</span>acom<span>&#46;</span>rwth-aachen<span>&#46;</span>de</a></p></li>
+</ul>
+<div class="toctree-wrapper compound">
+</div>
+</section>
+
+
+           </div>
+          </div>
+          <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer">
+        <a href="src.html" class="btn btn-neutral float-right" title="Source Code" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a>
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+
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+    <p>&#169; Copyright 2024, Jan Habscheid.</p>
+  </div>
+
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+<ul class="current">
+<li class="toctree-l1 current"><a class="current reference internal" href="#">Source Code</a><ul>
+<li class="toctree-l2"><a class="reference internal" href="#Eq04.create_refined_mesh"><code class="docutils literal notranslate"><span class="pre">create_refined_mesh()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#EqN.solve_System_Neq"><code class="docutils literal notranslate"><span class="pre">solve_System_Neq()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Eq04.solve_System_4eq"><code class="docutils literal notranslate"><span class="pre">solve_System_4eq()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Eq02.solve_System_2eq"><code class="docutils literal notranslate"><span class="pre">solve_System_2eq()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#ElectrolyticDiode.ElectrolyticDiode"><code class="docutils literal notranslate"><span class="pre">ElectrolyticDiode()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.C_dl"><code class="docutils literal notranslate"><span class="pre">C_dl()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.Phi_pot_center"><code class="docutils literal notranslate"><span class="pre">Phi_pot_center()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.Q_DL_dim_ana"><code class="docutils literal notranslate"><span class="pre">Q_DL_dim_ana()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.Q_DL_dimless_ana"><code class="docutils literal notranslate"><span class="pre">Q_DL_dimless_ana()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.Q_num_"><code class="docutils literal notranslate"><span class="pre">Q_num_()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.Q_num_dim"><code class="docutils literal notranslate"><span class="pre">Q_num_dim()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.dx"><code class="docutils literal notranslate"><span class="pre">dx()</span></code></a></li>
+<li class="toctree-l2"><a class="reference internal" href="#Helper_DoubleLayerCapacity.n"><code class="docutils literal notranslate"><span class="pre">n()</span></code></a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="example.html">Example</a></li>
+</ul>
+
+        </div>
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+             
+  <section id="source-code">
+<h1>Source Code<a class="headerlink" href="#source-code" title="Link to this heading"></a></h1>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>If the Newton solver diverges for any of the solutions, you may try to reduce the relaxation parameter.</p>
+</div>
+<dl class="py function">
+<dt class="sig sig-object py" id="Eq04.create_refined_mesh">
+<span class="sig-prename descclassname"><span class="pre">Eq04.</span></span><span class="sig-name descname"><span class="pre">create_refined_mesh</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">refinement_style</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">number_cells</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Mesh</span></span></span><a class="headerlink" href="#Eq04.create_refined_mesh" title="Link to this definition"></a></dt>
+<dd><p>Creates a one-dimensional mesh with a refined region at the left boundary</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>refinement_style</strong> (<em>str</em>) – How the mesh should be refined. Options are ‘log’, ‘hard_log’, ‘hard_hard_log’</p></li>
+<li><p><strong>number_cells</strong> (<em>int</em>) – Number of cells in the mesh</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>One-dimensional mesh, ready for use in FEniCSx</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>Mesh</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="EqN.solve_System_Neq">
+<span class="sig-prename descclassname"><span class="pre">EqN.</span></span><span class="sig-name descname"><span class="pre">solve_System_Neq</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phi_left</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_alpha</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">list</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">list</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">number_cells</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">PoissonBoltzmann</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">relax_param</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">refinement_style</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'uniform'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'Vector'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1e-08</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">500</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#EqN.solve_System_Neq" title="Link to this definition"></a></dt>
+<dd><p>Solve the dimensionless system of equations presented in: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model, B.Sc. Thesis Habscheid 2024</p>
+<dl>
+<dt>System of equations:</dt><dd><p>λ²Δ φ =−L²n^F</p>
+<p>a²∇p=−n^F∇ φ</p>
+<p>div(J_α)=0  α∈ {1,…,N−1}</p>
+</dd>
+</dl>
+<p>with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.</p>
+<p>! If the Newton solver diverges, you may try to reduce the relaxation parameter.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>phi_left</strong> (<em>float</em>) – Value of φ at the left boundary</p></li>
+<li><p><strong>phi_right</strong> (<em>float</em>) – Value of φ at the right boundary</p></li>
+<li><p><strong>p_right</strong> (<em>float</em>) – Value of p at the right boundary</p></li>
+<li><p><strong>z_alpha</strong> (<em>list</em>) – Charge numbers for species α = 1,…,N-1</p></li>
+<li><p><strong>y_R</strong> (<em>list</em>) – Atomic fractions at right boundary for species α = 1,…,N-1</p></li>
+<li><p><strong>K</strong> (<em>float</em><em> | </em><em>str</em>) – Dimensioness bulk modulus of the electrolyte. If ‘incompressible’, the system is solved for an incompressible electrolyte</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>number_cells</strong> (<em>int</em>) – Number of cells in the mesh</p></li>
+<li><p><strong>solvation</strong> (<em>float</em><em>, </em><em>optional</em>) – solvation number, not implemented yet, by default 0</p></li>
+<li><p><strong>PoissonBoltzmann</strong> (<em>bool</em><em>, </em><em>optional</em>) – Solve classical Nernst-Planck model with the use of the Poisson-Boltzmann formulation if True, else solve the presented model by Dreyer, Guhlke, Müller, Not implemented yet, by default False</p></li>
+<li><p><strong>relax_param</strong> (<em>float</em><em>, </em><em>optional</em>) – Relaxation parameter for the Newton solver
+xₙ₊₁ = γ xₙ f(xₙ)/f’(xₙ) with γ the relaxation parameter
+, by default None -&gt; Determined automatically</p></li>
+<li><p><strong>x0</strong> (<em>float</em><em>, </em><em>optional</em>) – Left boundary of the domain, by default 0</p></li>
+<li><p><strong>x1</strong> (<em>float</em><em>, </em><em>optional</em>) – Right boundary of the domain, by default 1</p></li>
+<li><p><strong>refinement_style</strong> (<em>str</em><em>, </em><em>optional</em>) – Specify for refinement towards zero
+Options are ‘uniform’, ‘log’, ‘hard_log’, ‘hard_hard_log’ by default ‘uniform’</p></li>
+<li><p><strong>return_type</strong> (<em>str</em><em>, </em><em>optional</em>) – ‘Vector’ or ‘Scalar’ (not implemented yet, should be implemented in a later version), ‘Scalar’ returns dolfinx.fem type and ‘Vector’ numpy arrays of the solution, by default ‘Vector’</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – Relative tolerance for Newton solver, by default 1e-8</p></li>
+<li><p><strong>max_iter</strong> (<em>float</em><em>, </em><em>optional</em>) – Maximum number of Newton iterations, by default 500</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Returns atomic fractions for species A and C, electric potential, pressure, and the mesh
+If return_type is ‘Vector’, the solution is returned as numpy arrays
+Only return_type ‘Vector’ is implemented yet</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>y_A, y_C, phi, p, msh</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Eq04.solve_System_4eq">
+<span class="sig-prename descclassname"><span class="pre">Eq04.</span></span><span class="sig-name descname"><span class="pre">solve_System_4eq</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phi_left</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_A_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">number_cells</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">PoissonBoltzmann</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">relax_param</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">refinement_style</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'uniform'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'Scalar'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1e-08</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">500</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#Eq04.solve_System_4eq" title="Link to this definition"></a></dt>
+<dd><p>Solve the dimensionless system of equations presented in: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model, B.Sc. Thesis Habscheid 2024</p>
+<dl>
+<dt>System of equations:</dt><dd><p>λ²Δ φ =−L²n^F</p>
+<p>a²∇p=−n^F∇ φ</p>
+<p>div(J_A)=0</p>
+<p>div(J_C)=0</p>
+</dd>
+</dl>
+<p>with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.</p>
+<p>! If the Newton solver diverges, you may try to reduce the relaxation parameter.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>phi_left</strong> (<em>float</em>) – Value of φ at the left boundary</p></li>
+<li><p><strong>phi_right</strong> (<em>float</em>) – Value of φ at the right boundary</p></li>
+<li><p><strong>p_right</strong> (<em>float</em>) – Value of p at the right boundary</p></li>
+<li><p><strong>z_A</strong> (<em>float</em>) – Charge number of species A</p></li>
+<li><p><strong>z_C</strong> (<em>float</em>) – Charge number of species C</p></li>
+<li><p><strong>y_A_R</strong> (<em>float</em>) – Atomic fractions of species A at right boundary</p></li>
+<li><p><strong>y_C_R</strong> (<em>float</em>) – Atomic fractions of species C at right boundary</p></li>
+<li><p><strong>K</strong> (<em>float</em><em> | </em><em>str</em>) – Dimensioness bulk modulus of the electrolyte. If ‘incompressible’, the system is solved for an incompressible electrolyte</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>number_cells</strong> (<em>int</em>) – Number of cells in the mesh</p></li>
+<li><p><strong>solvation</strong> (<em>float</em><em>, </em><em>optional</em>) – solvation number, by default 0</p></li>
+<li><p><strong>PoissonBoltzmann</strong> (<em>bool</em><em>, </em><em>optional</em>) – Solve classical Nernst-Planck model with the use of the Poisson-Boltzmann formulation if True, else solve the presented model by Dreyer, Guhlke, Müller, by default False</p></li>
+<li><p><strong>relax_param</strong> (<em>float</em><em>, </em><em>optional</em>) – Relaxation parameter for the Newton solver
+xₙ₊₁ = γ xₙ f(xₙ)/f’(xₙ) with γ the relaxation parameter
+, by default None -&gt; Determined automatically</p></li>
+<li><p><strong>x0</strong> (<em>float</em><em>, </em><em>optional</em>) – Left boundary of the domain, by default 0</p></li>
+<li><p><strong>x1</strong> (<em>float</em><em>, </em><em>optional</em>) – Right boundary of the domain, by default 1</p></li>
+<li><p><strong>refinement_style</strong> (<em>str</em><em>, </em><em>optional</em>) – Specify for refinement towards zero
+Options are ‘uniform’, ‘log’, ‘hard_log’, ‘hard_hard_log’ by default ‘uniform’</p></li>
+<li><p><strong>return_type</strong> (<em>str</em><em>, </em><em>optional</em>) – ‘Vector’ or ‘Scalar’, ‘Scalar’ returns dolfinx.fem type and ‘Vector’ numpy arrays of the solution, by default ‘Scalar’</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – Relative tolerance for Newton solver, by default 1e-8</p></li>
+<li><p><strong>max_iter</strong> (<em>float</em><em>, </em><em>optional</em>) – Maximum number of Newton iterations, by default 500</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Returns atomic fractions for species A and C, electric potential, pressure, and the mesh
+If return_type is ‘Vector’, the solution is returned as numpy arrays</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>y_A, y_C, phi, p, msh</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Eq02.solve_System_2eq">
+<span class="sig-prename descclassname"><span class="pre">Eq02.</span></span><span class="sig-name descname"><span class="pre">solve_System_2eq</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">phi_left</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_right</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_A_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">number_cells</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">relax_param</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x0</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x1</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">refinement_style</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'uniform'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'Scalar'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1e-08</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">500</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#Eq02.solve_System_2eq" title="Link to this definition"></a></dt>
+<dd><p>Solve the simplified dimensionless system of equations presented in: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model, B.Sc. Thesis Habscheid 2024</p>
+<dl>
+<dt>System of equations:</dt><dd><p>λ²Δ φ =−L²n^F</p>
+<p>a²∇p=−n^F∇ φ</p>
+</dd>
+<dt>with the space charge</dt><dd><p>n^F = z_A y_A(φ, p) + z_C y_C(φ ,p)</p>
+</dd>
+<dt>if the mixture is compressible:</dt><dd><p>y_alpha = C_alpha * (K+p−1)^(−κ+1)a²K exp(−z_α φ)</p>
+</dd>
+<dt>if the mixture is incompressible:</dt><dd><p>y_alpha = D_alpha * exp(−(κ+1)a²p−z_α φ)</p>
+</dd>
+</dl>
+<p>with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.</p>
+<p>! If the Newton solver diverges, you may try to reduce the relaxation parameter.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>phi_left</strong> (<em>float</em>) – Value of φ at the left boundary</p></li>
+<li><p><strong>phi_right</strong> (<em>float</em>) – Value of φ at the right boundary</p></li>
+<li><p><strong>p_right</strong> (<em>float</em>) – Value of p at the right boundary</p></li>
+<li><p><strong>z_A</strong> (<em>float</em>) – Charge number of species A</p></li>
+<li><p><strong>z_C</strong> (<em>float</em>) – Charge number of species C</p></li>
+<li><p><strong>y_A_R</strong> (<em>float</em>) – Atomic fractions of species A at right boundary</p></li>
+<li><p><strong>y_C_R</strong> (<em>float</em>) – Atomic fractions of species C at right boundary</p></li>
+<li><p><strong>K</strong> (<em>float</em><em> | </em><em>str</em>) – Dimensioness bulk modulus of the electrolyte. If ‘incompressible’, the system is solved for an incompressible electrolyte</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>number_cells</strong> (<em>int</em>) – Number of cells in the mesh</p></li>
+<li><p><strong>solvation</strong> (<em>float</em><em>, </em><em>optional</em>) – solvation number, by default 0</p></li>
+<li><p><strong>relax_param</strong> (<em>float</em><em>, </em><em>optional</em>) – Relaxation parameter for the Newton solver
+xₙ₊₁ = γ xₙ f(xₙ)/f’(xₙ) with γ the relaxation parameter
+, by default None -&gt; Determined automatically</p></li>
+<li><p><strong>x0</strong> (<em>float</em><em>, </em><em>optional</em>) – Left boundary of the domain, by default 0</p></li>
+<li><p><strong>x1</strong> (<em>float</em><em>, </em><em>optional</em>) – Right boundary of the domain, by default 1</p></li>
+<li><p><strong>refinement_style</strong> (<em>str</em><em>, </em><em>optional</em>) – Specify for refinement towards zero
+Options are ‘uniform’, ‘log’, ‘hard_log’, ‘hard_hard_log’ by default ‘uniform’</p></li>
+<li><p><strong>return_type</strong> (<em>str</em><em>, </em><em>optional</em>) – ‘Vector’ or ‘Scalar’, ‘Scalar’ returns dolfinx.fem type and ‘Vector’ numpy arrays of the solution, by default ‘Scalar’</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – Relative tolerance for Newton solver, by default 1e-8</p></li>
+<li><p><strong>max_iter</strong> (<em>float</em><em>, </em><em>optional</em>) – Maximum number of Newton iterations, by default 500</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Returns atomic fractions for species A and C, electric potential, pressure, and the mesh
+If return_type is ‘Vector’, the solution is returned as numpy arrays</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>y_A, y_C, phi, p, msh</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="ElectrolyticDiode.ElectrolyticDiode">
+<span class="sig-prename descclassname"><span class="pre">ElectrolyticDiode.</span></span><span class="sig-name descname"><span class="pre">ElectrolyticDiode</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Bias_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_bias</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">g_phi</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_A_bath</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C_bath</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">str</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">number_cells</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">list</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">PoissonBoltzmann</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">relax_param</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lx</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">2</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Ly</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">rtol</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1e-08</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_iter</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">500</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">return_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'Vector'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#ElectrolyticDiode.ElectrolyticDiode" title="Link to this definition"></a></dt>
+<dd><p>Solves the system of equations for the example of an electric diode</p>
+<p>Solve the dimensionless system of equations presented in: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model, B.Sc. Thesis Habscheid 2024</p>
+<p>! If the Newton solver diverges, you may try to reduce the relaxation parameter.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>Bias_type</strong> (<em>str</em>) – ForwardBias, NoBias, BackwardBias</p></li>
+<li><p><strong>phi_bias</strong> (<em>float</em>) – Bias in φ</p></li>
+<li><p><strong>g_phi</strong> (<em>float</em>) – Neumann boundary condition for φ</p></li>
+<li><p><strong>z_A</strong> (<em>float</em>) – Charge number of species A</p></li>
+<li><p><strong>z_C</strong> (<em>float</em>) – Charge number of species C</p></li>
+<li><p><strong>y_A_bath</strong> (<em>float</em>) – Bath concentration of anions</p></li>
+<li><p><strong>y_C_bath</strong> (<em>float</em>) – Bath concentration of cations</p></li>
+<li><p><strong>K</strong> (<em>float</em><em> | </em><em>str</em>) – Dimensioness bulk modulus of the electrolyte. If ‘incompressible’, the system is solved for an incompressible electrolyte
+! Only implemented for incompressible mixtures, yet</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>number_cells</strong> (<em>int</em>) – Number of cells in the mesh</p></li>
+<li><p><strong>solvation</strong> (<em>float</em><em>, </em><em>optional</em>) – solvation number, by default 0</p></li>
+<li><p><strong>PoissonBoltzmann</strong> (<em>bool</em><em>, </em><em>optional</em>) – Solve classical Nernst-Planck model with the use of the Poisson-Boltzmann formulation if True, else solve the presented model by Dreyer, Guhlke, Müller, by default False</p></li>
+<li><p><strong>relax_param</strong> (<em>float</em><em>, </em><em>optional</em>) – Relaxation parameter for the Newton solver
+xₙ₊₁ = γ xₙ f(xₙ)/f’(xₙ) with γ the relaxation parameter
+, by default None -&gt; Determined automatically</p></li>
+<li><p><strong>Lx</strong> (<em>float</em><em>, </em><em>optional</em>) – Length of domain in x-direction, by default 2</p></li>
+<li><p><strong>Ly</strong> (<em>float</em><em>, </em><em>optional</em>) – Length of domain in y-direction, by default 10</p></li>
+<li><p><strong>rtol</strong> (<em>float</em><em>, </em><em>optional</em>) – Relative tolerance for Newton solver, by default 1e-8</p></li>
+<li><p><strong>max_iter</strong> (<em>float</em><em>, </em><em>optional</em>) – Maximum number of Newton iterations, by default 500</p></li>
+<li><p><strong>return_type</strong> (<em>str</em><em>, </em><em>optional</em>) – Type of return value, by default ‘Vector’
+‘Vector’ -&gt; Returns the solution as a numpy array for triangle elements
+‘Extended’ -&gt; Returns the solution as both, a numpy array for triangle elements and the fenicsx function u</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Returns atomic fractions for species A and C, electric potential, pressure, and the mesh as numpy arrays for triangle elements. x_vector is a list of both dimensions [x, y]
+If return_type is ‘Extended’, also returns the fenicsx function u</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>y_A_vector, y_C_vector, phi_vector, p_vector, x_vector</p>
+</dd>
+</dl>
+</dd></dl>
+
+<p id="module-Helper_DoubleLayerCapacity">Jan Habscheid
+<a class="reference external" href="mailto:Jan&#46;Habscheid&#37;&#52;&#48;rwth-aachen&#46;de">Jan<span>&#46;</span>Habscheid<span>&#64;</span>rwth-aachen<span>&#46;</span>de</a></p>
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.C_dl">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">C_dl</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Q_DL</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Phi_pot</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">ndarray</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.C_dl" title="Link to this definition"></a></dt>
+<dd><p>Double Layer Capacity</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>Q_DL</strong> (<em>np.ndarray</em>) – Charge of the system</p></li>
+<li><p><strong>Phi_pot</strong> (<em>np.ndarray</em>) – Electric potential values</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Double Layer Capacity</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>np.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.Phi_pot_center">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">Phi_pot_center</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Phi_pot</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">ndarray</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.Phi_pot_center" title="Link to this definition"></a></dt>
+<dd><p>Returns vector with the center of the electric potential values.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>Phi_pot</strong> (<em>np.ndarray</em>) – Input vector with electric potential values.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Vector with the center of the electric potential values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>np.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.Q_DL_dim_ana">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">Q_DL_dim_ana</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y_A_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_N_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_L</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nR_m</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">e0</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">LR</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">float</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.Q_DL_dim_ana" title="Link to this definition"></a></dt>
+<dd><p>Calculates charge of the system using the analytical method in dimensionless units</p>
+<p>Q = sgn(φᴸ −φᴿ)λ√(2/(κ+1))(√(Λᴸ)−√(Λᴿ))</p>
+<p>with
+Λ = ln(∑_α D_α exp(−z_α φ + Eₚ a²)</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>y_A_R</strong> (<em>float</em>) – Value of anion fraction at the right boundary</p></li>
+<li><p><strong>y_C_R</strong> (<em>float</em>) – Value of cation fraction at the right boundary</p></li>
+<li><p><strong>y_N_R</strong> (<em>float</em>) – Value of neutral fraction at the right boundary</p></li>
+<li><p><strong>z_A</strong> (<em>float</em>) – Charge number of anions</p></li>
+<li><p><strong>z_C</strong> (<em>float</em>) – Charge number of cations</p></li>
+<li><p><strong>phi_L</strong> (<em>float</em>) – Value of electric potential at the left boundary</p></li>
+<li><p><strong>phi_R</strong> (<em>float</em>) – Value of electric potential at the right</p></li>
+<li><p><strong>p_R</strong> (<em>float</em>) – Value of pressure at the right boundary</p></li>
+<li><p><strong>K</strong> (<em>str</em><em> | </em><em>float</em>) – Bulk modulus, use ‘incompressible’ for an incompress</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>nR_m</strong> (<em>float</em>) – Reference number density in 1/m^3</p></li>
+<li><p><strong>e0</strong> (<em>float</em>) – Dielectric constant</p></li>
+<li><p><strong>LR</strong> (<em>float</em>) – Reference length in m</p></li>
+<li><p><strong>solvation</strong> (<em>float</em>) – Solvation number</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Charge of the system in µAs/cm³</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.Q_DL_dimless_ana">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">Q_DL_dimless_ana</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y_A_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_N_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_L</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">phi_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_R</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">Lambda2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">a2</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">solvation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">float</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.Q_DL_dimless_ana" title="Link to this definition"></a></dt>
+<dd><p>Calculates charge of the system using the analytical method in dimensionless units</p>
+<p>Q = sgn(φᴸ −φᴿ)λ√(2/(κ+1))(√(Λᴸ)−√(Λᴿ))</p>
+<p>with
+Λ = ln(∑_α D_α exp(−z_α φ + Eₚ a²)</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>y_A_R</strong> (<em>float</em>) – Value of anion fraction at the right boundary</p></li>
+<li><p><strong>y_C_R</strong> (<em>float</em>) – Value of cation fraction at the right boundary</p></li>
+<li><p><strong>y_N_R</strong> (<em>float</em>) – Value of neutral fraction at the right boundary</p></li>
+<li><p><strong>z_A</strong> (<em>float</em>) – Charge number of anions</p></li>
+<li><p><strong>z_C</strong> (<em>float</em>) – Charge number of cations</p></li>
+<li><p><strong>phi_L</strong> (<em>float</em>) – Electric potential at the left boundary [V]</p></li>
+<li><p><strong>phi_R</strong> (<em>float</em>) – Electric potential at the right boundary</p></li>
+<li><p><strong>p_R</strong> (<em>float</em>) – Pressure at the right boundary</p></li>
+<li><p><strong>K</strong> (<em>str</em><em> | </em><em>float</em>) – Bulk modulus, use ‘incompressible’ for an incompressible mixture</p></li>
+<li><p><strong>Lambda2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>a2</strong> (<em>float</em>) – Dimensionless parameter</p></li>
+<li><p><strong>solvation</strong> (<em>float</em>) – Solvation number</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Charge of the system in dimensionless units</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.Q_num_">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">Q_num_</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">-1.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">float</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.Q_num_" title="Link to this definition"></a></dt>
+<dd><p>Calculates the charge of the system</p>
+<p>Q = ∫_Ω n^F dΩ</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>y_A</strong> (<em>np.ndarray</em>) – Anion fraction</p></li>
+<li><p><strong>y_C</strong> (<em>np.ndarray</em>) – Cation fraction</p></li>
+<li><p><strong>n</strong> (<em>np.ndarray</em>) – Total number density</p></li>
+<li><p><strong>x</strong> (<em>np.ndarray</em>) – Spatial discretization</p></li>
+<li><p><strong>z_A</strong> (<em>float</em><em>, </em><em>optional</em>) – Charge number of anions, by default -1.0</p></li>
+<li><p><strong>z_C</strong> (<em>float</em><em>, </em><em>optional</em>) – Charge number of cations, by default 1.0</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Charge of the system</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.Q_num_dim">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">Q_num_dim</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">y_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_A</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">z_C</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nR_m</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">e0</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">LR</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">float</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.Q_num_dim" title="Link to this definition"></a></dt>
+<dd><p>Calculates the charge of the system</p>
+<p>Q = ∫_Ω n^F dΩ</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>y_A</strong> (<em>np.ndarray</em>) – Anion fraction</p></li>
+<li><p><strong>y_C</strong> (<em>np.ndarray</em>) – Cation fraction</p></li>
+<li><p><strong>n</strong> (<em>np.ndarray</em>) – Total number density</p></li>
+<li><p><strong>x</strong> (<em>np.ndarray</em>) – Spatial discretization</p></li>
+<li><p><strong>z_A</strong> (<em>float</em><em>, </em><em>optional</em>) – Charge number of anions, by default -1.0</p></li>
+<li><p><strong>z_C</strong> (<em>float</em><em>, </em><em>optional</em>) – Charge number of cations, by default 1.0</p></li>
+<li><p><strong>nR_m</strong> (<em>float</em>) – Reference number density in 1/m^3</p></li>
+<li><p><strong>e0</strong> (<em>float</em>) – Dielectric constant</p></li>
+<li><p><strong>LR</strong> (<em>float</em>) – Reference length in m</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Charge of the system</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.dx">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">dx</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">Phi_pot</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">float</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.dx" title="Link to this definition"></a></dt>
+<dd><p>Returns the difference between the first two electric potential values.</p>
+<p>Assumes that the electric potential values are equally spaced.</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><p><strong>Phi_pot</strong> (<em>np.ndarray</em>) – Input vector with electric potential values.</p>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Difference between the first two electric potential values.</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>float</p>
+</dd>
+</dl>
+</dd></dl>
+
+<dl class="py function">
+<dt class="sig sig-object py" id="Helper_DoubleLayerCapacity.n">
+<span class="sig-prename descclassname"><span class="pre">Helper_DoubleLayerCapacity.</span></span><span class="sig-name descname"><span class="pre">n</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">ndarray</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">K</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span><span class="w"> </span><span class="p"><span class="pre">|</span></span><span class="w"> </span><span class="pre">float</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">ndarray</span></span></span><a class="headerlink" href="#Helper_DoubleLayerCapacity.n" title="Link to this definition"></a></dt>
+<dd><p>Calculates the total number density</p>
+<dl class="field-list simple">
+<dt class="field-odd">Parameters<span class="colon">:</span></dt>
+<dd class="field-odd"><ul class="simple">
+<li><p><strong>p</strong> (<em>np.ndarray</em>) – Pressure</p></li>
+<li><p><strong>K</strong> (<em>str</em><em> | </em><em>float</em>) – Bulk modulus, use ‘incompressible’ for an incompressible mixture</p></li>
+</ul>
+</dd>
+<dt class="field-even">Returns<span class="colon">:</span></dt>
+<dd class="field-even"><p>Total number density</p>
+</dd>
+<dt class="field-odd">Return type<span class="colon">:</span></dt>
+<dd class="field-odd"><p>np.ndarray</p>
+</dd>
+</dl>
+</dd></dl>
+
+</section>
+
+
+           </div>
+          </div>
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+</html>
\ No newline at end of file
diff --git a/docs/make.bat b/docs/make.bat
new file mode 100644
index 0000000..747ffb7
--- /dev/null
+++ b/docs/make.bat
@@ -0,0 +1,35 @@
+@ECHO OFF
+
+pushd %~dp0
+
+REM Command file for Sphinx documentation
+
+if "%SPHINXBUILD%" == "" (
+	set SPHINXBUILD=sphinx-build
+)
+set SOURCEDIR=source
+set BUILDDIR=build
+
+%SPHINXBUILD% >NUL 2>NUL
+if errorlevel 9009 (
+	echo.
+	echo.The 'sphinx-build' command was not found. Make sure you have Sphinx
+	echo.installed, then set the SPHINXBUILD environment variable to point
+	echo.to the full path of the 'sphinx-build' executable. Alternatively you
+	echo.may add the Sphinx directory to PATH.
+	echo.
+	echo.If you don't have Sphinx installed, grab it from
+	echo.https://www.sphinx-doc.org/
+	exit /b 1
+)
+
+if "%1" == "" goto help
+
+%SPHINXBUILD% -M %1 %SOURCEDIR% %BUILDDIR% %SPHINXOPTS% %O%
+goto end
+
+:help
+%SPHINXBUILD% -M help %SOURCEDIR% %BUILDDIR% %SPHINXOPTS% %O%
+
+:end
+popd
diff --git a/docs/source/Figures/TernaryConcentrations.svg b/docs/source/Figures/TernaryConcentrations.svg
new file mode 100644
index 0000000..fc8a32d
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+ <defs>
+  <clipPath id="p04f71bc9be">
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+ </defs>
+</svg>
diff --git a/docs/source/conf.py b/docs/source/conf.py
new file mode 100644
index 0000000..4a4449d
--- /dev/null
+++ b/docs/source/conf.py
@@ -0,0 +1,46 @@
+# Configuration file for the Sphinx documentation builder.
+#
+# For the full list of built-in configuration values, see the documentation:
+# https://www.sphinx-doc.org/en/master/usage/configuration.html
+
+# -- 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'
+copyright = '2024, Jan Habscheid'
+author = 'Jan Habscheid'
+release = 'September, 2024'
+
+# Add src 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'))
+
+# -- General configuration ---------------------------------------------------
+# https://www.sphinx-doc.org/en/master/usage/configuration.html#general-configuration
+
+extensions = [
+              'sphinx.ext.napoleon',
+              'sphinx.ext.autodoc', 
+              'myst_parser',
+              'sphinx_copybutton',
+              'sphinx.ext.coverage',
+              'sphinx.ext.autosectionlabel',
+              ]
+
+templates_path = ['_templates']
+exclude_patterns = []
+
+source_suffix = {
+    '.rst': 'restructuredtext',
+    '.txt': 'markdown',
+    '.md': 'markdown',
+}
+
+autosummary_generate = False
+# autoclass_content = 'both'
+# -- Options for HTML output -------------------------------------------------
+# https://www.sphinx-doc.org/en/master/usage/configuration.html#options-for-html-output
+
+html_theme = 'sphinx_rtd_theme'
\ No newline at end of file
diff --git a/docs/source/example.rst b/docs/source/example.rst
new file mode 100644
index 0000000..7fd5139
--- /dev/null
+++ b/docs/source/example.rst
@@ -0,0 +1,105 @@
+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"
+
+| 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
+-----------------------------------------------------
+
+.. code-block:: python
+
+    import sys
+    import os
+
+    src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../..', 'src')
+    sys.path.insert(0, src_path)
+
+    from Eq04 import solve_System_4eq
+
+    del sys.path[0]
+
+    import matplotlib.pyplot as plt
+    import numpy as np
+
+Define parameters and boundary conditions
+-----------------------------------------
+
+.. code-block:: python
+
+    phi_left = 8.0
+    phi_right = 0.0
+    p_right = 0.0
+    y_A_R = 1/3
+    y_C_R = 1/3
+    z_A = -1.0
+    z_C = 1.0
+    K = 'incompressible'
+    Lambda2 = 8.553e-6
+    a2 = 7.5412e-4
+
+Define mesh and solver settings
+-------------------------------
+
+.. code-block:: python
+
+    number_cells = 1024
+    refinement_style = 'log'
+    rtol = 1e-8
+
+Solve the system
+----------------
+
+.. code-block:: python
+
+    y_A, y_C, phi, p, x = solve_System_4eq(phi_left, phi_right, p_right, z_A, z_C, y_A_R, y_C_R, K, Lambda2, a2, number_cells, relax_param=0.05, x0=0, x1=1,    refinement_style='hard_log', return_type='Vector', max_iter=1_000, rtol=rtol)
+
+Visualize the results
+---------------------
+
+.. code-block:: python
+
+    plt.figure()
+    plt.plot(x, phi)
+    plt.xlabel('x [-]')
+    plt.ylabel('$\\varphi$  [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.show()
+
+    plt.figure()
+    plt.plot(x, p)
+    plt.xlabel('x [-]')
+    plt.ylabel('$p$ [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.show()
+
+    plt.figure()
+    plt.plot(x, y_A, label='$y_A$')
+    plt.plot(x, y_C, label='$y_C$')
+    plt.plot(x, 1 - y_A - y_C, label='$y_S$')
+    plt.xlabel('x [-]')
+    plt.ylabel('$y_\\alpha$ [-]')
+    plt.xlim(0,0.05)
+    plt.grid()
+    plt.legend()
+    plt.show()
+
+The electric potential
+^^^^^^^^^^^^^^^^^^^^^^^
+
+.. image:: Figures/TernaryPotential.svg
+    :width: 600px
+
+The pressure 
+^^^^^^^^^^^^^
+
+.. image:: Figures/TernaryPressure.svg
+    :width: 600px
+
+The atomic fractions
+^^^^^^^^^^^^^^^^^^^^^
+.. image:: Figures/TernaryConcentrations.svg
+    :width: 600px
\ No newline at end of file
diff --git a/docs/source/index.rst b/docs/source/index.rst
new file mode 100644
index 0000000..0dc9c4b
--- /dev/null
+++ b/docs/source/index.rst
@@ -0,0 +1,82 @@
+.. 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.
+
+Reproducibility Repository for: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model documentation
+=====================================================================================================================
+
+.. admonition:: \ \
+
+   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: ToDo Add DOI
+
+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:
+
+- Insert link to thesis
+
+Abstract
+========
+
+| 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.
+
+| 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.
+
+| 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.
+
+.. 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
+
+Alternative installation
+========================
+| Use the "environment.yml" file to install all necessary environments
+
+.. code-block:: 
+
+   conda env create -f environment.yml
+
+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.
+
+Contact
+=======
+**Author**
+
+- Jan Habscheid
+- Jan.Habscheid@rwth-aachen.de
+
+**Supervisor**
+
+- Dr. Lambert Theissen
+- ACoM - Applied and Computational Mathematics
+- RWTH Aachen University
+- theisen@acom.rwth-aachen.de
+
+**Supervisor**
+
+- Prof. Dr. Manuel Torrilhon
+- ACoM - Applied and Computational Mathematics
+- RWTH Aachen University
+- mt@acom.rwth-aachen.de
+
+.. toctree::
+   src
+   example
+   :maxdepth: 2
+   :caption: Numerical Treatment of a Thermodynamically Consistent Electrolyte Model:
+   :hidden:
+
diff --git a/docs/source/src.rst b/docs/source/src.rst
new file mode 100644
index 0000000..f56f5d7
--- /dev/null
+++ b/docs/source/src.rst
@@ -0,0 +1,20 @@
+Source Code
+===========
+
+.. 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
+
+.. automodule:: Helper_DoubleLayerCapacity
+    :members:
+    :undoc-members:
+    :show-inheritance:
\ No newline at end of file
diff --git a/examples/ReproducableCode/TernaryElectrolyte.py b/examples/ReproducableCode/TernaryElectrolyte.py
index 85e7736..6d90ed9 100644
--- a/examples/ReproducableCode/TernaryElectrolyte.py
+++ b/examples/ReproducableCode/TernaryElectrolyte.py
@@ -41,7 +41,7 @@ rtol = 1e-8
 y_A, y_C, phi, p, x = solve_System_4eq(phi_left, phi_right, p_right, z_A, z_C, y_A_R, y_C_R, K, Lambda2, a2, number_cells, relax_param=0.05, x0=0, x1=1, 
 refinement_style='hard_log', return_type='Vector', max_iter=1_000, rtol=rtol)
 
-# Fine point, where space charge starts to diverge from zero
+# Find point, where space charge starts to diverge from zero
 index = 0
 nF = z_C * y_C + z_A * y_A
 for i, nF_ in enumerate(nF):
diff --git a/examples/ReproducableCode/temp.py b/examples/ReproducableCode/temp.py
new file mode 100644
index 0000000..f7b6ac0
--- /dev/null
+++ b/examples/ReproducableCode/temp.py
@@ -0,0 +1,66 @@
+import sys
+import os
+
+src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../..', 'src')
+sys.path.insert(0, src_path)
+
+from Eq04 import solve_System_4eq
+
+del sys.path[0]
+
+import matplotlib.pyplot as plt
+import numpy as np
+
+
+phi_left = 8.0
+phi_right = 0.0
+p_right = 0.0
+y_A_R = 1/3
+y_C_R = 1/3
+z_A = -1.0
+z_C = 1.0
+K = 'incompressible'
+Lambda2 = 8.553e-6
+a2 = 7.5412e-4
+
+
+
+number_cells = 1024
+refinement_style = 'log'
+rtol = 1e-8
+
+
+
+y_A, y_C, phi, p, x = solve_System_4eq(phi_left, phi_right, p_right, z_A, z_C, y_A_R, y_C_R, K, Lambda2, a2, number_cells, relax_param=0.05, x0=0, x1=1,    refinement_style='hard_log', return_type='Vector', max_iter=1_000, rtol=rtol)
+
+
+
+plt.figure()
+plt.plot(x, phi)
+plt.xlabel('x [-]')
+plt.ylabel('$\\varphi$  [-]')
+plt.xlim(0,0.05)
+plt.grid()
+plt.savefig('../../docs/source/Figures/TernaryPotential.svg')
+plt.show()
+
+plt.figure()
+plt.plot(x, p)
+plt.xlabel('x [-]')
+plt.ylabel('$p$ [-]')
+plt.xlim(0,0.05)
+plt.grid()
+plt.savefig('../../docs/source/Figures/TernaryPressure.svg')
+plt.show()
+
+plt.figure()
+plt.plot(x, y_A, label='$y_A$')
+plt.plot(x, y_C, label='$y_C$')
+plt.plot(x, 1 - y_A - y_C, label='$y_S$')
+plt.xlabel('x [-]')
+plt.ylabel('$y_\\alpha$ [-]')
+plt.xlim(0,0.05)
+plt.grid()
+plt.legend()
+plt.savefig('../../docs/source/Figures/TernaryConcentrations.svg')
+plt.show()
\ No newline at end of file
diff --git a/src/Eq02.py b/src/Eq02.py
index 6fd5135..0ec56e7 100644
--- a/src/Eq02.py
+++ b/src/Eq02.py
@@ -59,14 +59,17 @@ def solve_System_2eq(phi_left:float, phi_right:float, p_right:float, z_A:float,
 
     System of equations:
         λ²Δ φ =−L²n^F
+        
         a²∇p=−n^F∇ φ
         
-        with:
-            n^F = z_A y_A(φ, p) + z_C y_C(φ ,p)
-            if compressible:
-                y_alpha = C_alpha * (K+p−1)⁽−κ+1)a²K exp(−z_α φ)
-            if incompressible:
-                y_alpha = D_alpha * exp(−(κ+1)a²p−z_α φ)
+    with the space charge
+        n^F = z_A y_A(φ, p) + z_C y_C(φ ,p)
+
+    if the mixture is compressible: 
+        y_alpha = C_alpha * (K+p−1)^(−κ+1)a²K exp(−z_α φ)
+
+    if the mixture is incompressible: 
+        y_alpha = D_alpha * exp(−(κ+1)a²p−z_α φ)
 
     with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.
 
diff --git a/src/Eq04.py b/src/Eq04.py
index 360623f..4856d84 100644
--- a/src/Eq04.py
+++ b/src/Eq04.py
@@ -59,8 +59,12 @@ def solve_System_4eq(phi_left:float, phi_right:float, p_right:float, z_A:float,
 
     System of equations:
         λ²Δ φ =−L²n^F
+
         a²∇p=−n^F∇ φ
-        div(J_α)=0  α∈ {1,...,N−1}
+
+        div(J_A)=0
+
+        div(J_C)=0
 
     with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.
 
diff --git a/src/EqN.py b/src/EqN.py
index 7a278b6..9b48ed1 100644
--- a/src/EqN.py
+++ b/src/EqN.py
@@ -59,7 +59,9 @@ def solve_System_Neq(phi_left:float, phi_right:float, p_right:float, z_alpha:lis
 
     System of equations:
         λ²Δ φ =−L²n^F
+
         a²∇p=−n^F∇ φ
+        
         div(J_α)=0  α∈ {1,...,N−1}
 
     with φ the electric potential, p the pressure, n^F the total free charge density, J_α the diffusion fluxes of species α, λ² a dimensionless parameter, L²=1, a² a dimensionless parameter, N the number of species, and α the species index.
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delta 4616
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z45;E%i787gD#*xG(PV<CbD6x|M@2643n;oIKq<}t7JGbrN`7*D{4LhJ(%gc|B9qBK
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diff --git a/src/__pycache__/Eq02.cpython-312.pyc b/src/__pycache__/Eq02.cpython-312.pyc
index db7936e73aa933adfe164ec1a59346d55a066edb..a7b1e8dbb7236b5a8a7a2c061358e882cbd48424 100644
GIT binary patch
delta 490
zcmZo`V0>85dzzP*i-CcGVa3I3Y33Vw-FO-OCWr7kZ$8JX!Oti$`Mb==$@gWo#mX~F
zG89TOQWc5|5|dLEk~0#E(o?x6OUUV|aVaP$C}gIAWpgttN=l1T6*7wzlJj#5ic*V<
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uCyqH8!O3FIM{rzN+*#nt;#raqoFbNTF$B+@)ra*n*drfOHgnqjWdr~-;GO{h

delta 548
zcmaD@-`2p$dzzP*i-CcG;o;+>X{sA}-FO*WCx`Glo26DH<`(3n7V8J478K>DlqM%8
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h#=yeUQFWPF@&Uh4gU1a4;Reqee1Z*|f7?D|1OU<arF#GX

diff --git a/src/__pycache__/Eq04.cpython-312.pyc b/src/__pycache__/Eq04.cpython-312.pyc
index 4a7d8b92f39e7eaec7e53b5e695142678bd60f58..7466f34eeeaac5ca4a9999d68fc19efc50dbd4e1 100644
GIT binary patch
delta 564
zcmZ2CpK<<tM&8rByj%<n3=H8Hucdw6$ScO{?5CfbSE8S+A6lGRRIHzsn3s{L?~<Qf
znwy$eQmh|Pl%HOdn46oKm#&|lS)!j*oUEIglbT#ol%G>slA4>JlA2Szxsdm*7~`wS
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Vg@b{Gr=#jJv*d%#>wW(-0s!G=xM%<X

delta 558
zcmbO~pK;xMM&8rByj%<n3=Dk*SJS?4<Q3x;iPX=?&rQ|OO3c%D$xklLP0cGQp6tUb
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diff --git a/src/__pycache__/EqN.cpython-312.pyc b/src/__pycache__/EqN.cpython-312.pyc
index 77b8a2b5334718de5034fe97547302ddcb16794d..7e9ba8d9a74cf05b5623f83970ca5ae78c9a2285 100644
GIT binary patch
delta 562
zcmey_&A73fk@qw&FBby?1B3R(YiX?;dF^?fef4wmO7xTULyJ?3iuJP+^D+|kUGkGl
zb5rw5iuD7E^3#hFb8|EE()H6bOZ1b9lXX*bQj<%H@^dOnQgicDQgezopWuBf#ArHs
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delta 649
zcmdnk&G@sMk@qw&FBby?14BvI#kA^;y!O2HZTdy!B^hbO`bAax$vLIPC8<UF8Tq-X
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Mu1d@xI?^=)0KK`+^Z)<=

-- 
GitLab