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<project name="dragion" default="dist" basedir=".">
<description>
TODO
</description>
<property name="src" location="src"/>
<property name="build" location="build"/>
<property name="dist" location="dist"/>
<target name="init">
<tstamp/>
<mkdir dir="${build}"/>
</target>
<target name="compile" depends="init">
<javac destdir="${build}">
<src path="../mowgli/src"/>
<src path="../lestat/src"/>
<src path="../dragion/src"/>
<classpath>
<pathelement path="lib/antlr4-runtime-4.7.2/antlr4-runtime-4.7.2.jar"/>
<pathelement path="lib/commons-cli-1.5.0/commons-cli-1.5.0.jar"/>
<pathelement path="lib/commons-io-2.11.0/commons-io-2.11.0.jar"/>
<pathelement path="lib/commons-math3-3.6.1/commons-math3-3.6.1.jar"/>
<pathelement path="lib/guava-28.1/guava-28.1-jre.jar"/>
<pathelement path="lib/jfreechart-1.5.3/jfreechart-1.5.3.jar"/>
<pathelement path="lib/jna-5.4.0/jna-5.4.0.jar"/>
<pathelement path="lib/tomlj-1.0.0/tomlj-1.0.0.jar"/>
<pathelement path="${user.home}/local/vtk-8.2/lib64/vtk.jar"/>
</classpath>
</javac>
</target>
<target name="dist" depends="compile">
<mkdir dir="${dist}/lib"/>
<jar jarfile="${dist}/lib/dragion-${DSTAMP}.jar" basedir="${build}"/>
</target>
<target name="clean">
<delete dir="${build}"/>
<delete dir="${dist}"/>
</target>
</project>

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other

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# 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)

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# Sphinx build info version 1
# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
config: 2add6e08e92b233349ce3e889c403cb8
tags: 645f666f9bcd5a90fca523b33c5a78b7

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Bipolar ions: uniform source and loss (unfinished)
==================================================
Problem Statement
-----------------
The bipolar balance equation (in the absence of aerosol particles) is
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}
where :math:`n_\pm` are respective ion concentrations and :math:`\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}` [Harrison and Tammet]. Sans aerosols, :math:`n_+=n_-`.
**dragion** can be tested against this bipolar balance equation with uniform ``VolumeSource`` and ``CollisionalSinkBipolar`` instances for each ion species.
The implementation is forthcoming, to be based on ``dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java``.
The ``VolumeSource`` objects uniformly distribute positive and negative ions across the simulation ``Box`` at a steady rate, representing the constant source term :math:`q`.
The ``CollisionalSinkBipolar`` algorithm simulates the respective loss terms :math:`- \alpha n_+n_-` by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing :math:`n_+n_-` in each grid cell
(2) removing :math:`\alpha \: n_+n_- \: \mathrm{d}t \: \mathrm{d}^3r \: / np2c` macroparticles of each species from the grid cell (leaving zero if exceeding existing number)
We would expect the simulated :math:`\frac{\mathrm{d}n}{\mathrm{d}t}` to equal :math:`0`, integrated over many timesteps; in other words, the simulated concentration should be constant and the positive and negative concentrations equal.
Numerical Validation
--------------------
Taking :math:`q = 4\times10^6 \mathrm{m}^{-3} \mathrm{s}^{-1}` and :math:`\frac{\mathrm{d}n_+}{\mathrm{d}t}=0` yields a steady-state concentration of :math:`n_{equil}=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG].
TODO

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Unipolar relaxation
===================
Problem Statement
-----------------
Consider an initially uncharged conducting sphere placed in a uniform, cold, unipolar ion bath.
After a while we bias the sphere to a polarity opposite that of the ions, and then immediately disconnect it, leaving the sphere electrically floating at a nonzero potential.
The resulting unipolar current can be derived from the electric field
.. math::
:nowrap:
\begin{eqnarray}
E_r=\frac{Q}{4 \pi \varepsilon_0 r^2}
\end{eqnarray}
and the definition of mobility
.. math::
:nowrap:
\begin{eqnarray}
\vec{v} = \mu \vec{E}
\end{eqnarray}
to get
.. math::
:nowrap:
\begin{eqnarray}
I = n q v_r A = n q \cdot \mu E_r \cdot 4 \pi R^2
\end{eqnarray}
where :math:`n` is ion concentration, :math:`q` is ion charge, and :math:`R` is radius of the sphere. This yields the differential equation for charge on the sphere
.. math::
:nowrap:
\begin{eqnarray}
\dot{Q}=n q \mu \cdot \frac{Q}{4 \pi \varepsilon_0 R^2} \cdot 4 \pi R^2=Q \frac{n q \mu}{\varepsilon_0}
\end{eqnarray}
Converting :math:`Q` to :math:`V` using the definition of capacitance
.. math::
:nowrap:
\begin{eqnarray}
C V = Q
\end{eqnarray}
and noting that
.. math::
:nowrap:
\begin{eqnarray}
\sigma = n |q| \mu
\end{eqnarray}
the solution is
.. math::
:nowrap:
\begin{eqnarray}
V(t) = V_0 \exp\left(-\frac{t}{\varepsilon_0/\sigma}\right)
\end{eqnarray}
The negative sign in the exponential comes from the sign of :math:`q` relative to :math:`V_0`.
Numerical Validation
--------------------
Runs simulating the relaxation phenomenon for a conducting sphere can be found in ``dragion/runs/sphere``.
Initially, positive unipolar ions are loaded uniformly into the domain, with concentration :math:`n=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG], and a sphere of radius :math:`R=0.5 \mathrm{m}` is biased to :math:`-10 \mathrm{V}` then left floating.
Note: this simulation does not include a uniform background source or collisional sink.
The resulting time-progression is illustrated by a sequence of 3D screenshots:
- :math:`t=0 \mathrm{s}`
.. image:: relaxation-sphere-3d_0.png
- :math:`t=500 \mathrm{s}`
.. image:: relaxation-sphere-3d_1.png
- :math:`t=1000 \mathrm{s}`
.. image:: relaxation-sphere-3d_2.png
As the sphere collects nearby ions its (negative) charge level and electric field strength diminish.
The voltage drop is exponential, and its slope (in log-voltage) can be used to back out an estimate of atmospheric conductivity, :math:`\sigma = \mathrm{slope} \cdot \varepsilon_0`.
Here is a plot of (log10) voltage over time:
.. image:: relaxation-sphere.png
Notice that overall the **dragion** sphere relaxes more slowly than predicted by theory.
However, there is a slight inflection in the simulated voltage at :math:`t=1200 \mathrm{s}`.
A comparison of slopes before and after this knee, and the full end-to-end slope provide estimates for conductivity, using :math:`\sigma = \mathrm{slope} \cdot \varepsilon_0`:
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
| Fit | Slope (relaxation rate) :math:`[\mathrm{s}^{-1}]` | :math:`\sigma [\mathrm{S}/\mathrm{m}]` |
+=====================================+====================================================+==================================================+
| End-to-end | -0.0011 | :math:`0.95\times10^{-14}` |
| | | |
| Before :math:`t=1200 \: \mathrm{s}` | -0.0009 | :math:`0.79\times10^{-14}` |
| | | |
| After :math:`t=1200 \:\mathrm{s}` | -0.0013 | :math:`1.2\times10^{-14}` |
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
| Theoretical | -0.0016 | :math:`1.4\times10^{-14}` |
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
*More discussion is needed. Why is there an inflection point? It is probably related to domain shape and size. Why is the late-time estimate the best?*
At very late times the near-surface ion concentration will be depleted, so without a background source charging will eventually halt. In this regime the assumption of steady near-surface concentration would be violated and the theoretical model breaks down.

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Unipolar ions: uniform source and loss
======================================
Problem Statement
-----------------
The bipolar balance equation (in the absence of aerosol particles) is
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}
where :math:`n_\pm` are respective ion concentrations and :math:`\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}` [Harrison and Tammet]. Sans aerosols, :math:`n_+=n_-`, so that a single species reads
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n}{\mathrm{d}t} = q - \alpha n^2
\end{eqnarray}
**dragion** can be tested against this unipolar balance equation with a uniform ``VolumeSource`` and ``CollisionalSinkUnipolar``; the implementation is in ``dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java``.
The ``VolumeSource`` uniformly distributes particles across the simulation ``Box`` at a steady rate, representing the constant source term :math:`q`.
The ``CollisionalSinkUnipolar`` algorithm simulates the loss term :math:`- \alpha n^2` by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing :math:`n^2` in each grid cell
(2) removing :math:`\alpha \: n^2 \: \mathrm{d}t \: \mathrm{d}^3r \: / np2c` macroparticles from each grid cell (leaving zero if exceeding existing number)
We would expect the simulated :math:`\frac{\mathrm{d}n}{\mathrm{d}t}` to equal :math:`0`, integrated over many timesteps; in other words, the simulated concentration should be constant.
Numerical Validation
--------------------
Taking :math:`q = 4\times10^6 \mathrm{m}^{-3} \mathrm{s}^{-1}` and :math:`\frac{\mathrm{d}n_+}{\mathrm{d}t}=0` yields a steady-state unipolar concentration of :math:`n_{equil}=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG].
The ``SourceSinkTest`` starts with an empty 5 m simulation box. As time progresses the ``VolumeSource`` fills the domain with macroparticles, until the "real" concentration reaches :math:`n_{equil}`.
Crucial simulation parameters are the macroparticle "size", controlled by :math:`np2c` and the grid spacing used for particle-particle collisions. For this numerical experiment the collision grid cell-size is the domain length divided by 64.
At :math:`np2c=10^7` the mean concentration peaks rapidly but then unphysically falls off: the number of macroparticles per grid-cell is far too low, with cell size / particle size ratio :math:=`(5 \mathrm{m}/64)^3/(n_{equil}/np2c) = 0.1`. Reducing :math:`np2c` by two orders of 10 increases the # particles per grid cell to :math:`~10`, and the expected steady-state concentration is maintained.
.. image:: coll-equil.png

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Getting started
===============
Dependencies
------------
VTK
^^^^^^^
You will need to compile [VTK 8.2](vtk.org/download) from source, with Java bindings enabled. Typically on MacOS or Linux, using [CMake](https://cmake.org/), this looks like::
wget https://www.vtk.org/files/release/8.2/VTK-8.2.0.tar.gz
tar xf VTK-8.2.0.tar.gz
cd VTK-8.2.0
mkdir build
cd build
cmake -DCMAKE_INSTALL_PREFIX=$VTK_HOME -DVTK_WRAP_JAVA=TRUE ..
make install
Note that ``$VTK_HOME`` must be defined prior to invocation of cmake. The VTK Java bindings jar-file ``vtk.jar`` and a multitude of dynamically linked sub-libraries are installed into ``$VTK_HOME/lib``.
mowgli
^^^^^^
Implementation of surfaces, distribution samplers, and other low-level building blocks are currently part of the **mowgli** library. Clone it from <http://hardin.jhuapl.edu:8080/scm/zimmemi1/mowgli>. Further documentation can be found *here*.
lestat
^^^^^^
TODO: copy from lestat getting-started docs
Installation and test run
-------------------------
Get the code from <http://hardin.jhuapl.edu:8080/scm/zimmemi1/dragion>. Assume that it is cloned into `~/dragion`.
For now the classpath and runtime library path are passed explicitly to the JRE from the command line (this will probably chang>
To print a helpful message, try::
DYLD_LIBRARY_PATH=~/local/vtk-8.2/lib \
java -Djava.library.path=/Users/zimmemi1/local/vtk-8.2/lib \
-classpath ~/dragion/bin:~/dragion/lib/commons-math3-3.6.1/commons-math3-3.6.1.jar:~/dragion/lib/guava-28.1/guava-28.1->
dragion.Main

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.. dragion documentation master file, created by
sphinx-quickstart on Thu Aug 25 16:19:38 2022.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
dragion
=======
A Java API for modeling electrostatic interactions between atmospheric ions and charged surfaces, including:
- arbitrarily shaped homogeneous dielectrics and (biased or floating) conductors
- self-consistent surface charging through ion collection
- ion diffusion and mobility (in the presence of external and surface fields)
- volumetric ion sources and collisional loss process
.. toctree::
:maxdepth: 2
:caption: Contents:
getstarted.rst
physvalid.rst
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`

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Physics validation
==================
.. toctree::
:maxdepth: 1
examples/unipolar_source_loss.rst
examples/bipolar_source_loss.rst
examples/relaxation_sphere.rst

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}
table.footnote + table.footnote {
margin-top: -15px;
border-top: none;
}
table.field-list th {
padding: 0 0.8em 0 0;
}
table.field-list td {
padding: 0;
}
table.field-list p {
margin-bottom: 0.8em;
}
/* Cloned from
* https://github.com/sphinx-doc/sphinx/commit/ef60dbfce09286b20b7385333d63a60321784e68
*/
.field-name {
-moz-hyphens: manual;
-ms-hyphens: manual;
-webkit-hyphens: manual;
hyphens: manual;
}
table.footnote td.label {
width: .1px;
padding: 0.3em 0 0.3em 0.5em;
}
table.footnote td {
padding: 0.3em 0.5em;
}
dl {
margin: 0;
padding: 0;
}
dl dd {
margin-left: 30px;
}
blockquote {
margin: 0 0 0 30px;
padding: 0;
}
ul, ol {
/* Matches the 30px from the narrow-screen "li > ul" selector below */
margin: 10px 0 10px 30px;
padding: 0;
}
pre {
background: #EEE;
padding: 7px 30px;
margin: 15px 0px;
line-height: 1.3em;
}
div.viewcode-block:target {
background: #ffd;
}
dl pre, blockquote pre, li pre {
margin-left: 0;
padding-left: 30px;
}
tt, code {
background-color: #ecf0f3;
color: #222;
/* padding: 1px 2px; */
}
tt.xref, code.xref, a tt {
background-color: #FBFBFB;
border-bottom: 1px solid #fff;
}
a.reference {
text-decoration: none;
border-bottom: 1px dotted #004B6B;
}
/* Don't put an underline on images */
a.image-reference, a.image-reference:hover {
border-bottom: none;
}
a.reference:hover {
border-bottom: 1px solid #6D4100;
}
a.footnote-reference {
text-decoration: none;
font-size: 0.7em;
vertical-align: top;
border-bottom: 1px dotted #004B6B;
}
a.footnote-reference:hover {
border-bottom: 1px solid #6D4100;
}
a:hover tt, a:hover code {
background: #EEE;
}
@media screen and (max-width: 870px) {
div.sphinxsidebar {
display: none;
}
div.document {
width: 100%;
}
div.documentwrapper {
margin-left: 0;
margin-top: 0;
margin-right: 0;
margin-bottom: 0;
}
div.bodywrapper {
margin-top: 0;
margin-right: 0;
margin-bottom: 0;
margin-left: 0;
}
ul {
margin-left: 0;
}
li > ul {
/* Matches the 30px from the "ul, ol" selector above */
margin-left: 30px;
}
.document {
width: auto;
}
.footer {
width: auto;
}
.bodywrapper {
margin: 0;
}
.footer {
width: auto;
}
.github {
display: none;
}
}
@media screen and (max-width: 875px) {
body {
margin: 0;
padding: 20px 30px;
}
div.documentwrapper {
float: none;
background: #fff;
}
div.sphinxsidebar {
display: block;
float: none;
width: 102.5%;
margin: 50px -30px -20px -30px;
padding: 10px 20px;
background: #333;
color: #FFF;
}
div.sphinxsidebar h3, div.sphinxsidebar h4, div.sphinxsidebar p,
div.sphinxsidebar h3 a {
color: #fff;
}
div.sphinxsidebar a {
color: #AAA;
}
div.sphinxsidebar p.logo {
display: none;
}
div.document {
width: 100%;
margin: 0;
}
div.footer {
display: none;
}
div.bodywrapper {
margin: 0;
}
div.body {
min-height: 0;
padding: 0;
}
.rtd_doc_footer {
display: none;
}
.document {
width: auto;
}
.footer {
width: auto;
}
.footer {
width: auto;
}
.github {
display: none;
}
}
/* misc. */
.revsys-inline {
display: none!important;
}
/* Make nested-list/multi-paragraph items look better in Releases changelog
* pages. Without this, docutils' magical list fuckery causes inconsistent
* formatting between different release sub-lists.
*/
div#changelog > div.section > ul > li > p:only-child {
margin-bottom: 0;
}
/* Hide fugly table cell borders in ..bibliography:: directive output */
table.docutils.citation, table.docutils.citation td, table.docutils.citation th {
border: none;
/* Below needed in some edge cases; if not applied, bottom shadows appear */
-moz-box-shadow: none;
-webkit-box-shadow: none;
box-shadow: none;
}
/* relbar */
.related {
line-height: 30px;
width: 100%;
font-size: 0.9rem;
}
.related.top {
border-bottom: 1px solid #EEE;
margin-bottom: 20px;
}
.related.bottom {
border-top: 1px solid #EEE;
}
.related ul {
padding: 0;
margin: 0;
list-style: none;
}
.related li {
display: inline;
}
nav#rellinks {
float: right;
}
nav#rellinks li+li:before {
content: "|";
}
nav#breadcrumbs li+li:before {
content: "\00BB";
}
/* Hide certain items when printing */
@media print {
div.related {
display: none;
}
}

906
doc/sphinx/build/html/_static/basic.css Normal file
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/*
* basic.css
* ~~~~~~~~~
*
* Sphinx stylesheet -- basic theme.
*
* :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
* :license: BSD, see LICENSE for details.
*
*/
/* -- main layout ----------------------------------------------------------- */
div.clearer {
clear: both;
}
div.section::after {
display: block;
content: '';
clear: left;
}
/* -- relbar ---------------------------------------------------------------- */
div.related {
width: 100%;
font-size: 90%;
}
div.related h3 {
display: none;
}
div.related ul {
margin: 0;
padding: 0 0 0 10px;
list-style: none;
}
div.related li {
display: inline;
}
div.related li.right {
float: right;
margin-right: 5px;
}
/* -- sidebar --------------------------------------------------------------- */
div.sphinxsidebarwrapper {
padding: 10px 5px 0 10px;
}
div.sphinxsidebar {
float: left;
width: 230px;
margin-left: -100%;
font-size: 90%;
word-wrap: break-word;
overflow-wrap : break-word;
}
div.sphinxsidebar ul {
list-style: none;
}
div.sphinxsidebar ul ul,
div.sphinxsidebar ul.want-points {
margin-left: 20px;
list-style: square;
}
div.sphinxsidebar ul ul {
margin-top: 0;
margin-bottom: 0;
}
div.sphinxsidebar form {
margin-top: 10px;
}
div.sphinxsidebar input {
border: 1px solid #98dbcc;
font-family: sans-serif;
font-size: 1em;
}
div.sphinxsidebar #searchbox form.search {
overflow: hidden;
}
div.sphinxsidebar #searchbox input[type="text"] {
float: left;
width: 80%;
padding: 0.25em;
box-sizing: border-box;
}
div.sphinxsidebar #searchbox input[type="submit"] {
float: left;
width: 20%;
border-left: none;
padding: 0.25em;
box-sizing: border-box;
}
img {
border: 0;
max-width: 100%;
}
/* -- search page ----------------------------------------------------------- */
ul.search {
margin: 10px 0 0 20px;
padding: 0;
}
ul.search li {
padding: 5px 0 5px 20px;
background-image: url(file.png);
background-repeat: no-repeat;
background-position: 0 7px;
}
ul.search li a {
font-weight: bold;
}
ul.search li p.context {
color: #888;
margin: 2px 0 0 30px;
text-align: left;
}
ul.keywordmatches li.goodmatch a {
font-weight: bold;
}
/* -- index page ------------------------------------------------------------ */
table.contentstable {
width: 90%;
margin-left: auto;
margin-right: auto;
}
table.contentstable p.biglink {
line-height: 150%;
}
a.biglink {
font-size: 1.3em;
}
span.linkdescr {
font-style: italic;
padding-top: 5px;
font-size: 90%;
}
/* -- general index --------------------------------------------------------- */
table.indextable {
width: 100%;
}
table.indextable td {
text-align: left;
vertical-align: top;
}
table.indextable ul {
margin-top: 0;
margin-bottom: 0;
list-style-type: none;
}
table.indextable > tbody > tr > td > ul {
padding-left: 0em;
}
table.indextable tr.pcap {
height: 10px;
}
table.indextable tr.cap {
margin-top: 10px;
background-color: #f2f2f2;
}
img.toggler {
margin-right: 3px;
margin-top: 3px;
cursor: pointer;
}
div.modindex-jumpbox {
border-top: 1px solid #ddd;
border-bottom: 1px solid #ddd;
margin: 1em 0 1em 0;
padding: 0.4em;
}
div.genindex-jumpbox {
border-top: 1px solid #ddd;
border-bottom: 1px solid #ddd;
margin: 1em 0 1em 0;
padding: 0.4em;
}
/* -- domain module index --------------------------------------------------- */
table.modindextable td {
padding: 2px;
border-collapse: collapse;
}
/* -- general body styles --------------------------------------------------- */
div.body {
min-width: 450px;
max-width: 800px;
}
div.body p, div.body dd, div.body li, div.body blockquote {
-moz-hyphens: auto;
-ms-hyphens: auto;
-webkit-hyphens: auto;
hyphens: auto;
}
a.headerlink {
visibility: hidden;
}
a.brackets:before,
span.brackets > a:before{
content: "[";
}
a.brackets:after,
span.brackets > a:after {
content: "]";
}
h1:hover > a.headerlink,
h2:hover > a.headerlink,
h3:hover > a.headerlink,
h4:hover > a.headerlink,
h5:hover > a.headerlink,
h6:hover > a.headerlink,
dt:hover > a.headerlink,
caption:hover > a.headerlink,
p.caption:hover > a.headerlink,
div.code-block-caption:hover > a.headerlink {
visibility: visible;
}
div.body p.caption {
text-align: inherit;
}
div.body td {
text-align: left;
}
.first {
margin-top: 0 !important;
}
p.rubric {
margin-top: 30px;
font-weight: bold;
}
img.align-left, figure.align-left, .figure.align-left, object.align-left {
clear: left;
float: left;
margin-right: 1em;
}
img.align-right, figure.align-right, .figure.align-right, object.align-right {
clear: right;
float: right;
margin-left: 1em;
}
img.align-center, figure.align-center, .figure.align-center, object.align-center {
display: block;
margin-left: auto;
margin-right: auto;
}
img.align-default, figure.align-default, .figure.align-default {
display: block;
margin-left: auto;
margin-right: auto;
}
.align-left {
text-align: left;
}
.align-center {
text-align: center;
}
.align-default {
text-align: center;
}
.align-right {
text-align: right;
}
/* -- sidebars -------------------------------------------------------------- */
div.sidebar,
aside.sidebar {
margin: 0 0 0.5em 1em;
border: 1px solid #ddb;
padding: 7px;
background-color: #ffe;
width: 40%;
float: right;
clear: right;
overflow-x: auto;
}
p.sidebar-title {
font-weight: bold;
}
div.admonition, div.topic, blockquote {
clear: left;
}
/* -- topics ---------------------------------------------------------------- */
div.topic {
border: 1px solid #ccc;
padding: 7px;
margin: 10px 0 10px 0;
}
p.topic-title {
font-size: 1.1em;
font-weight: bold;
margin-top: 10px;
}
/* -- admonitions ----------------------------------------------------------- */
div.admonition {
margin-top: 10px;
margin-bottom: 10px;
padding: 7px;
}
div.admonition dt {
font-weight: bold;
}
p.admonition-title {
margin: 0px 10px 5px 0px;
font-weight: bold;
}
div.body p.centered {
text-align: center;
margin-top: 25px;
}
/* -- content of sidebars/topics/admonitions -------------------------------- */
div.sidebar > :last-child,
aside.sidebar > :last-child,
div.topic > :last-child,
div.admonition > :last-child {
margin-bottom: 0;
}
div.sidebar::after,
aside.sidebar::after,
div.topic::after,
div.admonition::after,
blockquote::after {
display: block;
content: '';
clear: both;
}
/* -- tables ---------------------------------------------------------------- */
table.docutils {
margin-top: 10px;
margin-bottom: 10px;
border: 0;
border-collapse: collapse;
}
table.align-center {
margin-left: auto;
margin-right: auto;
}
table.align-default {
margin-left: auto;
margin-right: auto;
}
table caption span.caption-number {
font-style: italic;
}
table caption span.caption-text {
}
table.docutils td, table.docutils th {
padding: 1px 8px 1px 5px;
border-top: 0;
border-left: 0;
border-right: 0;
border-bottom: 1px solid #aaa;
}
table.footnote td, table.footnote th {
border: 0 !important;
}
th {
text-align: left;
padding-right: 5px;
}
table.citation {
border-left: solid 1px gray;
margin-left: 1px;
}
table.citation td {
border-bottom: none;
}
th > :first-child,
td > :first-child {
margin-top: 0px;
}
th > :last-child,
td > :last-child {
margin-bottom: 0px;
}
/* -- figures --------------------------------------------------------------- */
div.figure, figure {
margin: 0.5em;
padding: 0.5em;
}
div.figure p.caption, figcaption {
padding: 0.3em;
}
div.figure p.caption span.caption-number,
figcaption span.caption-number {
font-style: italic;
}
div.figure p.caption span.caption-text,
figcaption span.caption-text {
}
/* -- field list styles ----------------------------------------------------- */
table.field-list td, table.field-list th {
border: 0 !important;
}
.field-list ul {
margin: 0;
padding-left: 1em;
}
.field-list p {
margin: 0;
}
.field-name {
-moz-hyphens: manual;
-ms-hyphens: manual;
-webkit-hyphens: manual;
hyphens: manual;
}
/* -- hlist styles ---------------------------------------------------------- */
table.hlist {
margin: 1em 0;
}
table.hlist td {
vertical-align: top;
}
/* -- object description styles --------------------------------------------- */
.sig {
font-family: 'Consolas', 'Menlo', 'DejaVu Sans Mono', 'Bitstream Vera Sans Mono', monospace;
}
.sig-name, code.descname {
background-color: transparent;
font-weight: bold;
}
.sig-name {
font-size: 1.1em;
}
code.descname {
font-size: 1.2em;
}
.sig-prename, code.descclassname {
background-color: transparent;
}
.optional {
font-size: 1.3em;
}
.sig-paren {
font-size: larger;
}
.sig-param.n {
font-style: italic;
}
/* C++ specific styling */
.sig-inline.c-texpr,
.sig-inline.cpp-texpr {
font-family: unset;
}
.sig.c .k, .sig.c .kt,
.sig.cpp .k, .sig.cpp .kt {
color: #0033B3;
}
.sig.c .m,
.sig.cpp .m {
color: #1750EB;
}
.sig.c .s, .sig.c .sc,
.sig.cpp .s, .sig.cpp .sc {
color: #067D17;
}
/* -- other body styles ----------------------------------------------------- */
ol.arabic {
list-style: decimal;
}
ol.loweralpha {
list-style: lower-alpha;
}
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;
}
dl.footnote > dt,
dl.citation > dt {
float: left;
margin-right: 0.5em;
}
dl.footnote > dd,
dl.citation > dd {
margin-bottom: 0em;
}
dl.footnote > dd:after,
dl.citation > dd: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 > dt:after {
content: ":";
}
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;
}
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;
}
/* -- 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;
}
}

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/* This file intentionally left blank. */

358
doc/sphinx/build/html/_static/doctools.js Normal file
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/*
* doctools.js
* ~~~~~~~~~~~
*
* Sphinx JavaScript utilities for all documentation.
*
* :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS.
* :license: BSD, see LICENSE for details.
*
*/
/**
* select a different prefix for underscore
*/
$u = _.noConflict();
/**
* make the code below compatible with browsers without
* an installed firebug like debugger
if (!window.console || !console.firebug) {
var names = ["log", "debug", "info", "warn", "error", "assert", "dir",
"dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace",
"profile", "profileEnd"];
window.console = {};
for (var i = 0; i < names.length; ++i)
window.console[names[i]] = function() {};
}
*/
/**
* 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;
/**
* This function returns the parsed url parameters of the
* current request. Multiple values per key are supported,
* it will always return arrays of strings for the value parts.
*/
jQuery.getQueryParameters = function(s) {
if (typeof s === 'undefined')
s = document.location.search;
var parts = s.substr(s.indexOf('?') + 1).split('&');
var result = {};
for (var i = 0; i < parts.length; i++) {
var tmp = parts[i].split('=', 2);
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) &&
!jQuery(node.parentNode).hasClass("nohighlight")) {
var span;
var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg");
if (isInSVG) {
span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
} else {
span = document.createElement("span");
span.className = className;
}
span.appendChild(document.createTextNode(val.substr(pos, text.length)));
node.parentNode.insertBefore(span, node.parentNode.insertBefore(
document.createTextNode(val.substr(pos + text.length)),
node.nextSibling));
node.nodeValue = val.substr(0, pos);
if (isInSVG) {
var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect");
var bbox = node.parentElement.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": node.parentNode,
"target": rect});
}
}
}
else if (!jQuery(node).is("button, select, textarea")) {
jQuery.each(node.childNodes, function() {
highlight(this, addItems);
});
}
}
var addItems = [];
var result = this.each(function() {
highlight(this, addItems);
});
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 {
browser: match[ 1 ] || "",
version: match[ 2 ] || "0"
};
};
jQuery.browser = {};
jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true;
}
/**
* Small JavaScript module for the documentation.
*/
var Documentation = {
init : function() {
this.fixFirefoxAnchorBug();
this.highlightSearchWords();
this.initIndexTable();
this.initOnKeyListeners();
},
/**
* i18n support
*/
TRANSLATIONS : {},
PLURAL_EXPR : function(n) { return 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 : function(string) {
var translated = Documentation.TRANSLATIONS[string];
if (typeof translated === 'undefined')
return string;
return (typeof translated === 'string') ? translated : translated[0];
},
ngettext : function(singular, plural, n) {
var translated = Documentation.TRANSLATIONS[singular];
if (typeof translated === 'undefined')
return (n == 1) ? singular : plural;
return translated[Documentation.PLURALEXPR(n)];
},
addTranslations : function(catalog) {
for (var key in catalog.messages)
this.TRANSLATIONS[key] = catalog.messages[key];
this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')');
this.LOCALE = catalog.locale;
},
/**
* add context elements like header anchor links
*/
addContextElements : function() {
$('div[id] > :header:first').each(function() {
$('<a class="headerlink">\u00B6</a>').
attr('href', '#' + this.id).
attr('title', _('Permalink to this headline')).
appendTo(this);
});
$('dt[id]').each(function() {
$('<a class="headerlink">\u00B6</a>').
attr('href', '#' + this.id).
attr('title', _('Permalink to this definition')).
appendTo(this);
});
},
/**
* workaround a firefox stupidity
* see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075
*/
fixFirefoxAnchorBug : function() {
if (document.location.hash && $.browser.mozilla)
window.setTimeout(function() {
document.location.href += '';
}, 10);
},
/**
* highlight the search words provided in the url in the text
*/
highlightSearchWords : function() {
var params = $.getQueryParameters();
var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : [];
if (terms.length) {
var body = $('div.body');
if (!body.length) {
body = $('body');
}
window.setTimeout(function() {
$.each(terms, function() {
body.highlightText(this.toLowerCase(), 'highlighted');
});
}, 10);
$('<p class="highlight-link"><a href="javascript:Documentation.' +
'hideSearchWords()">' + _('Hide Search Matches') + '</a></p>')
.appendTo($('#searchbox'));
}
},
/**
* init the domain index toggle buttons
*/
initIndexTable : function() {
var togglers = $('img.toggler').click(function() {
var src = $(this).attr('src');
var idnum = $(this).attr('id').substr(7);
$('tr.cg-' + idnum).toggle();
if (src.substr(-9) === 'minus.png')
$(this).attr('src', src.substr(0, src.length-9) + 'plus.png');
else
$(this).attr('src', src.substr(0, src.length-8) + 'minus.png');
}).css('display', '');
if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) {
togglers.click();
}
},
/**
* helper function to hide the search marks again
*/
hideSearchWords : function() {
$('#searchbox .highlight-link').fadeOut(300);
$('span.highlighted').removeClass('highlighted');
var url = new URL(window.location);
url.searchParams.delete('highlight');
window.history.replaceState({}, '', url);
},
/**
* helper function to focus on search bar
*/
focusSearchBar : function() {
$('input[name=q]').first().focus();
},
/**
* make the url absolute
*/
makeURL : function(relativeURL) {
return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL;
},
/**
* get the current relative url
*/
getCurrentURL : function() {
var path = document.location.pathname;
var parts = path.split(/\//);
$.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() {
if (this === '..')
parts.pop();
});
var url = parts.join('/');
return path.substring(url.lastIndexOf('/') + 1, path.length - 1);
},
initOnKeyListeners: function() {
// only install a listener if it is really needed
if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS &&
!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
return;
$(document).keydown(function(event) {
var activeElementType = document.activeElement.tagName;
// don't navigate when in search box, textarea, dropdown or button
if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT'
&& activeElementType !== 'BUTTON') {
if (event.altKey || event.ctrlKey || event.metaKey)
return;
if (!event.shiftKey) {
switch (event.key) {
case 'ArrowLeft':
if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS)
break;
var prevHref = $('link[rel="prev"]').prop('href');
if (prevHref) {
window.location.href = prevHref;
return false;
}
break;
case 'ArrowRight':
if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS)
break;
var nextHref = $('link[rel="next"]').prop('href');
if (nextHref) {
window.location.href = nextHref;
return false;
}
break;
case 'Escape':
if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
break;
Documentation.hideSearchWords();
return false;
}
}
// some keyboard layouts may need Shift to get /
switch (event.key) {
case '/':
if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS)
break;
Documentation.focusSearchBar();
return false;
}
}
});
}
};
// quick alias for translations
_ = Documentation.gettext;
$(document).ready(function() {
Documentation.init();
});

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var DOCUMENTATION_OPTIONS = {
URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'),
VERSION: '',
LANGUAGE: 'None',
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,
};

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/*
* 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 2007-2022 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, is 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;
}
}
var splitChars = (function() {
var result = {};
var singles = [96, 180, 187, 191, 215, 247, 749, 885, 903, 907, 909, 930, 1014, 1648,
1748, 1809, 2416, 2473, 2481, 2526, 2601, 2609, 2612, 2615, 2653, 2702,
2706, 2729, 2737, 2740, 2857, 2865, 2868, 2910, 2928, 2948, 2961, 2971,
2973, 3085, 3089, 3113, 3124, 3213, 3217, 3241, 3252, 3295, 3341, 3345,
3369, 3506, 3516, 3633, 3715, 3721, 3736, 3744, 3748, 3750, 3756, 3761,
3781, 3912, 4239, 4347, 4681, 4695, 4697, 4745, 4785, 4799, 4801, 4823,
4881, 5760, 5901, 5997, 6313, 7405, 8024, 8026, 8028, 8030, 8117, 8125,
8133, 8181, 8468, 8485, 8487, 8489, 8494, 8527, 11311, 11359, 11687, 11695,
11703, 11711, 11719, 11727, 11735, 12448, 12539, 43010, 43014, 43019, 43587,
43696, 43713, 64286, 64297, 64311, 64317, 64319, 64322, 64325, 65141];
var i, j, start, end;
for (i = 0; i < singles.length; i++) {
result[singles[i]] = true;
}
var ranges = [[0, 47], [58, 64], [91, 94], [123, 169], [171, 177], [182, 184], [706, 709],
[722, 735], [741, 747], [751, 879], [888, 889], [894, 901], [1154, 1161],
[1318, 1328], [1367, 1368], [1370, 1376], [1416, 1487], [1515, 1519], [1523, 1568],
[1611, 1631], [1642, 1645], [1750, 1764], [1767, 1773], [1789, 1790], [1792, 1807],
[1840, 1868], [1958, 1968], [1970, 1983], [2027, 2035], [2038, 2041], [2043, 2047],
[2070, 2073], [2075, 2083], [2085, 2087], [2089, 2307], [2362, 2364], [2366, 2383],
[2385, 2391], [2402, 2405], [2419, 2424], [2432, 2436], [2445, 2446], [2449, 2450],
[2483, 2485], [2490, 2492], [2494, 2509], [2511, 2523], [2530, 2533], [2546, 2547],
[2554, 2564], [2571, 2574], [2577, 2578], [2618, 2648], [2655, 2661], [2672, 2673],
[2677, 2692], [2746, 2748], [2750, 2767], [2769, 2783], [2786, 2789], [2800, 2820],
[2829, 2830], [2833, 2834], [2874, 2876], [2878, 2907], [2914, 2917], [2930, 2946],
[2955, 2957], [2966, 2968], [2976, 2978], [2981, 2983], [2987, 2989], [3002, 3023],
[3025, 3045], [3059, 3076], [3130, 3132], [3134, 3159], [3162, 3167], [3170, 3173],
[3184, 3191], [3199, 3204], [3258, 3260], [3262, 3293], [3298, 3301], [3312, 3332],
[3386, 3388], [3390, 3423], [3426, 3429], [3446, 3449], [3456, 3460], [3479, 3481],
[3518, 3519], [3527, 3584], [3636, 3647], [3655, 3663], [3674, 3712], [3717, 3718],
[3723, 3724], [3726, 3731], [3752, 3753], [3764, 3772], [3774, 3775], [3783, 3791],
[3802, 3803], [3806, 3839], [3841, 3871], [3892, 3903], [3949, 3975], [3980, 4095],
[4139, 4158], [4170, 4175], [4182, 4185], [4190, 4192], [4194, 4196], [4199, 4205],
[4209, 4212], [4226, 4237], [4250, 4255], [4294, 4303], [4349, 4351], [4686, 4687],
[4702, 4703], [4750, 4751], [4790, 4791], [4806, 4807], [4886, 4887], [4955, 4968],
[4989, 4991], [5008, 5023], [5109, 5120], [5741, 5742], [5787, 5791], [5867, 5869],
[5873, 5887], [5906, 5919], [5938, 5951], [5970, 5983], [6001, 6015], [6068, 6102],
[6104, 6107], [6109, 6111], [6122, 6127], [6138, 6159], [6170, 6175], [6264, 6271],
[6315, 6319], [6390, 6399], [6429, 6469], [6510, 6511], [6517, 6527], [6572, 6592],
[6600, 6607], [6619, 6655], [6679, 6687], [6741, 6783], [6794, 6799], [6810, 6822],
[6824, 6916], [6964, 6980], [6988, 6991], [7002, 7042], [7073, 7085], [7098, 7167],
[7204, 7231], [7242, 7244], [7294, 7400], [7410, 7423], [7616, 7679], [7958, 7959],
[7966, 7967], [8006, 8007], [8014, 8015], [8062, 8063], [8127, 8129], [8141, 8143],
[8148, 8149], [8156, 8159], [8173, 8177], [8189, 8303], [8306, 8307], [8314, 8318],
[8330, 8335], [8341, 8449], [8451, 8454], [8456, 8457], [8470, 8472], [8478, 8483],
[8506, 8507], [8512, 8516], [8522, 8525], [8586, 9311], [9372, 9449], [9472, 10101],
[10132, 11263], [11493, 11498], [11503, 11516], [11518, 11519], [11558, 11567],
[11622, 11630], [11632, 11647], [11671, 11679], [11743, 11822], [11824, 12292],
[12296, 12320], [12330, 12336], [12342, 12343], [12349, 12352], [12439, 12444],
[12544, 12548], [12590, 12592], [12687, 12689], [12694, 12703], [12728, 12783],
[12800, 12831], [12842, 12880], [12896, 12927], [12938, 12976], [12992, 13311],
[19894, 19967], [40908, 40959], [42125, 42191], [42238, 42239], [42509, 42511],
[42540, 42559], [42592, 42593], [42607, 42622], [42648, 42655], [42736, 42774],
[42784, 42785], [42889, 42890], [42893, 43002], [43043, 43055], [43062, 43071],
[43124, 43137], [43188, 43215], [43226, 43249], [43256, 43258], [43260, 43263],
[43302, 43311], [43335, 43359], [43389, 43395], [43443, 43470], [43482, 43519],
[43561, 43583], [43596, 43599], [43610, 43615], [43639, 43641], [43643, 43647],
[43698, 43700], [43703, 43704], [43710, 43711], [43715, 43738], [43742, 43967],
[44003, 44015], [44026, 44031], [55204, 55215], [55239, 55242], [55292, 55295],
[57344, 63743], [64046, 64047], [64110, 64111], [64218, 64255], [64263, 64274],
[64280, 64284], [64434, 64466], [64830, 64847], [64912, 64913], [64968, 65007],
[65020, 65135], [65277, 65295], [65306, 65312], [65339, 65344], [65371, 65381],
[65471, 65473], [65480, 65481], [65488, 65489], [65496, 65497]];
for (i = 0; i < ranges.length; i++) {
start = ranges[i][0];
end = ranges[i][1];
for (j = start; j <= end; j++) {
result[j] = true;
}
}
return result;
})();
function splitQuery(query) {
var result = [];
var start = -1;
for (var i = 0; i < query.length; i++) {
if (splitChars[query.charCodeAt(i)]) {
if (start !== -1) {
result.push(query.slice(start, i));
start = -1;
}
} else if (start === -1) {
start = i;
}
}
if (start !== -1) {
result.push(query.slice(start));
}
return result;
}

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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: #8f5902; font-style: italic } /* Comment */
.highlight .err { color: #a40000; border: 1px solid #ef2929 } /* Error */
.highlight .g { color: #000000 } /* Generic */
.highlight .k { color: #004461; font-weight: bold } /* Keyword */
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<section id="bipolar-ions-uniform-source-and-loss-unfinished">
<h1>Bipolar ions: uniform source and loss (unfinished)<a class="headerlink" href="#bipolar-ions-uniform-source-and-loss-unfinished" title="Permalink to this headline"></a></h1>
<section id="problem-statement">
<h2>Problem Statement<a class="headerlink" href="#problem-statement" title="Permalink to this headline"></a></h2>
<p>The bipolar balance equation (in the absence of aerosol particles) is</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}</div><p>where <span class="math notranslate nohighlight">\(n_\pm\)</span> are respective ion concentrations and <span class="math notranslate nohighlight">\(\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}\)</span> [Harrison and Tammet]. Sans aerosols, <span class="math notranslate nohighlight">\(n_+=n_-\)</span>.</p>
<p><strong>dragion</strong> can be tested against this bipolar balance equation with uniform <code class="docutils literal notranslate"><span class="pre">VolumeSource</span></code> and <code class="docutils literal notranslate"><span class="pre">CollisionalSinkBipolar</span></code> instances for each ion species.</p>
<p>The implementation is forthcoming, to be based on <code class="docutils literal notranslate"><span class="pre">dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java</span></code>.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">VolumeSource</span></code> objects uniformly distribute positive and negative ions across the simulation <code class="docutils literal notranslate"><span class="pre">Box</span></code> at a steady rate, representing the constant source term <span class="math notranslate nohighlight">\(q\)</span>.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">CollisionalSinkBipolar</span></code> algorithm simulates the respective loss terms <span class="math notranslate nohighlight">\(- \alpha n_+n_-\)</span> by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing <span class="math notranslate nohighlight">\(n_+n_-\)</span> in each grid cell
(2) removing <span class="math notranslate nohighlight">\(\alpha \: n_+n_- \: \mathrm{d}t \: \mathrm{d}^3r \: / np2c\)</span> macroparticles of each species from the grid cell (leaving zero if exceeding existing number)</p>
<p>We would expect the simulated <span class="math notranslate nohighlight">\(\frac{\mathrm{d}n}{\mathrm{d}t}\)</span> to equal <span class="math notranslate nohighlight">\(0\)</span>, integrated over many timesteps; in other words, the simulated concentration should be constant and the positive and negative concentrations equal.</p>
</section>
<section id="numerical-validation">
<h2>Numerical Validation<a class="headerlink" href="#numerical-validation" title="Permalink to this headline"></a></h2>
<p>Taking <span class="math notranslate nohighlight">\(q = 4\times10^6 \mathrm{m}^{-3} \mathrm{s}^{-1}\)</span> and <span class="math notranslate nohighlight">\(\frac{\mathrm{d}n_+}{\mathrm{d}t}=0\)</span> yields a steady-state concentration of <span class="math notranslate nohighlight">\(n_{equil}=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}\)</span> [Lorenz, MMRTG].</p>
<p>TODO</p>
</section>
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<section id="unipolar-relaxation">
<h1>Unipolar relaxation<a class="headerlink" href="#unipolar-relaxation" title="Permalink to this headline"></a></h1>
<section id="problem-statement">
<h2>Problem Statement<a class="headerlink" href="#problem-statement" title="Permalink to this headline"></a></h2>
<p>Consider an initially uncharged conducting sphere placed in a uniform, cold, unipolar ion bath.
After a while we bias the sphere to a polarity opposite that of the ions, and then immediately disconnect it, leaving the sphere electrically floating at a nonzero potential.
The resulting unipolar current can be derived from the electric field</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
E_r=\frac{Q}{4 \pi \varepsilon_0 r^2}
\end{eqnarray}</div><p>and the definition of mobility</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\vec{v} = \mu \vec{E}
\end{eqnarray}</div><p>to get</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
I = n q v_r A = n q \cdot \mu E_r \cdot 4 \pi R^2
\end{eqnarray}</div><p>where <span class="math notranslate nohighlight">\(n\)</span> is ion concentration, <span class="math notranslate nohighlight">\(q\)</span> is ion charge, and <span class="math notranslate nohighlight">\(R\)</span> is radius of the sphere. This yields the differential equation for charge on the sphere</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\dot{Q}=n q \mu \cdot \frac{Q}{4 \pi \varepsilon_0 R^2} \cdot 4 \pi R^2=Q \frac{n q \mu}{\varepsilon_0}
\end{eqnarray}</div><p>Converting <span class="math notranslate nohighlight">\(Q\)</span> to <span class="math notranslate nohighlight">\(V\)</span> using the definition of capacitance</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
C V = Q
\end{eqnarray}</div><p>and noting that</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\sigma = n |q| \mu
\end{eqnarray}</div><p>the solution is</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
V(t) = V_0 \exp\left(-\frac{t}{\varepsilon_0/\sigma}\right)
\end{eqnarray}</div><p>The negative sign in the exponential comes from the sign of <span class="math notranslate nohighlight">\(q\)</span> relative to <span class="math notranslate nohighlight">\(V_0\)</span>.</p>
</section>
<section id="numerical-validation">
<h2>Numerical Validation<a class="headerlink" href="#numerical-validation" title="Permalink to this headline"></a></h2>
<p>Runs simulating the relaxation phenomenon for a conducting sphere can be found in <code class="docutils literal notranslate"><span class="pre">dragion/runs/sphere</span></code>.</p>
<p>Initially, positive unipolar ions are loaded uniformly into the domain, with concentration <span class="math notranslate nohighlight">\(n=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}\)</span> [Lorenz, MMRTG], and a sphere of radius <span class="math notranslate nohighlight">\(R=0.5 \mathrm{m}\)</span> is biased to <span class="math notranslate nohighlight">\(-10 \mathrm{V}\)</span> then left floating.
Note: this simulation does not include a uniform background source or collisional sink.</p>
<p>The resulting time-progression is illustrated by a sequence of 3D screenshots:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(t=0 \mathrm{s}\)</span></p></li>
</ul>
<img alt="../_images/relaxation-sphere-3d_0.png" src="../_images/relaxation-sphere-3d_0.png" />
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(t=500 \mathrm{s}\)</span></p></li>
</ul>
<img alt="../_images/relaxation-sphere-3d_1.png" src="../_images/relaxation-sphere-3d_1.png" />
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(t=1000 \mathrm{s}\)</span></p></li>
</ul>
<img alt="../_images/relaxation-sphere-3d_2.png" src="../_images/relaxation-sphere-3d_2.png" />
<p>As the sphere collects nearby ions its (negative) charge level and electric field strength diminish.
The voltage drop is exponential, and its slope (in log-voltage) can be used to back out an estimate of atmospheric conductivity, <span class="math notranslate nohighlight">\(\sigma = \mathrm{slope} \cdot \varepsilon_0\)</span>.</p>
<p>Here is a plot of (log10) voltage over time:</p>
<img alt="../_images/relaxation-sphere.png" src="../_images/relaxation-sphere.png" />
<p>Notice that overall the <strong>dragion</strong> sphere relaxes more slowly than predicted by theory.
However, there is a slight inflection in the simulated voltage at <span class="math notranslate nohighlight">\(t=1200 \mathrm{s}\)</span>.
A comparison of slopes before and after this knee, and the full end-to-end slope provide estimates for conductivity, using <span class="math notranslate nohighlight">\(\sigma = \mathrm{slope} \cdot \varepsilon_0\)</span>:</p>
<table class="docutils align-default">
<colgroup>
<col style="width: 27%" />
<col style="width: 37%" />
<col style="width: 36%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"><p>Fit</p></th>
<th class="head"><p>Slope (relaxation rate) <span class="math notranslate nohighlight">\([\mathrm{s}^{-1}]\)</span></p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(\sigma [\mathrm{S}/\mathrm{m}]\)</span></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>End-to-end</p>
<p>Before <span class="math notranslate nohighlight">\(t=1200 \: \mathrm{s}\)</span></p>
<p>After <span class="math notranslate nohighlight">\(t=1200 \:\mathrm{s}\)</span></p>
</td>
<td><p>-0.0011</p>
<p>-0.0009</p>
<p>-0.0013</p>
</td>
<td><p><span class="math notranslate nohighlight">\(0.95\times10^{-14}\)</span></p>
<p><span class="math notranslate nohighlight">\(0.79\times10^{-14}\)</span></p>
<p><span class="math notranslate nohighlight">\(1.2\times10^{-14}\)</span></p>
</td>
</tr>
<tr class="row-odd"><td><p>Theoretical</p></td>
<td><p>-0.0016</p></td>
<td><p><span class="math notranslate nohighlight">\(1.4\times10^{-14}\)</span></p></td>
</tr>
</tbody>
</table>
<p><em>More discussion is needed. Why is there an inflection point? It is probably related to domain shape and size. Why is the late-time estimate the best?</em></p>
<p>At very late times the near-surface ion concentration will be depleted, so without a background source charging will eventually halt. In this regime the assumption of steady near-surface concentration would be violated and the theoretical model breaks down.</p>
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<section id="unipolar-ions-uniform-source-and-loss">
<h1>Unipolar ions: uniform source and loss<a class="headerlink" href="#unipolar-ions-uniform-source-and-loss" title="Permalink to this headline"></a></h1>
<section id="problem-statement">
<h2>Problem Statement<a class="headerlink" href="#problem-statement" title="Permalink to this headline"></a></h2>
<p>The bipolar balance equation (in the absence of aerosol particles) is</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}</div><p>where <span class="math notranslate nohighlight">\(n_\pm\)</span> are respective ion concentrations and <span class="math notranslate nohighlight">\(\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}\)</span> [Harrison and Tammet]. Sans aerosols, <span class="math notranslate nohighlight">\(n_+=n_-\)</span>, so that a single species reads</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\frac{\mathrm{d}n}{\mathrm{d}t} = q - \alpha n^2
\end{eqnarray}</div><p><strong>dragion</strong> can be tested against this unipolar balance equation with a uniform <code class="docutils literal notranslate"><span class="pre">VolumeSource</span></code> and <code class="docutils literal notranslate"><span class="pre">CollisionalSinkUnipolar</span></code>; the implementation is in <code class="docutils literal notranslate"><span class="pre">dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java</span></code>.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">VolumeSource</span></code> uniformly distributes particles across the simulation <code class="docutils literal notranslate"><span class="pre">Box</span></code> at a steady rate, representing the constant source term <span class="math notranslate nohighlight">\(q\)</span>.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">CollisionalSinkUnipolar</span></code> algorithm simulates the loss term <span class="math notranslate nohighlight">\(- \alpha n^2\)</span> by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing <span class="math notranslate nohighlight">\(n^2\)</span> in each grid cell
(2) removing <span class="math notranslate nohighlight">\(\alpha \: n^2 \: \mathrm{d}t \: \mathrm{d}^3r \: / np2c\)</span> macroparticles from each grid cell (leaving zero if exceeding existing number)</p>
<p>We would expect the simulated <span class="math notranslate nohighlight">\(\frac{\mathrm{d}n}{\mathrm{d}t}\)</span> to equal <span class="math notranslate nohighlight">\(0\)</span>, integrated over many timesteps; in other words, the simulated concentration should be constant.</p>
</section>
<section id="numerical-validation">
<h2>Numerical Validation<a class="headerlink" href="#numerical-validation" title="Permalink to this headline"></a></h2>
<p>Taking <span class="math notranslate nohighlight">\(q = 4\times10^6 \mathrm{m}^{-3} \mathrm{s}^{-1}\)</span> and <span class="math notranslate nohighlight">\(\frac{\mathrm{d}n_+}{\mathrm{d}t}=0\)</span> yields a steady-state unipolar concentration of <span class="math notranslate nohighlight">\(n_{equil}=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}\)</span> [Lorenz, MMRTG].</p>
<p>The <code class="docutils literal notranslate"><span class="pre">SourceSinkTest</span></code> starts with an empty 5 m simulation box. As time progresses the <code class="docutils literal notranslate"><span class="pre">VolumeSource</span></code> fills the domain with macroparticles, until the “real” concentration reaches <span class="math notranslate nohighlight">\(n_{equil}\)</span>.</p>
<p>Crucial simulation parameters are the macroparticle “size”, controlled by <span class="math notranslate nohighlight">\(np2c\)</span> and the grid spacing used for particle-particle collisions. For this numerical experiment the collision grid cell-size is the domain length divided by 64.
At <span class="math notranslate nohighlight">\(np2c=10^7\)</span> the mean concentration peaks rapidly but then unphysically falls off: the number of macroparticles per grid-cell is far too low, with cell size / particle size ratio :math:=`(5 mathrm{m}/64)^3/(n_{equil}/np2c) = 0.1`. Reducing <span class="math notranslate nohighlight">\(np2c\)</span> by two orders of 10 increases the # particles per grid cell to <span class="math notranslate nohighlight">\(~10\)</span>, and the expected steady-state concentration is maintained.</p>
<img alt="../_images/coll-equil.png" src="../_images/coll-equil.png" />
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<section id="dependencies">
<h2>Dependencies<a class="headerlink" href="#dependencies" title="Permalink to this headline"></a></h2>
<section id="vtk">
<h3>VTK<a class="headerlink" href="#vtk" title="Permalink to this headline"></a></h3>
<p>You will need to compile [VTK 8.2](vtk.org/download) from source, with Java bindings enabled. Typically on MacOS or Linux, using [CMake](<a class="reference external" href="https://cmake.org/">https://cmake.org/</a>), this looks like:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>wget https://www.vtk.org/files/release/8.2/VTK-8.2.0.tar.gz
tar xf VTK-8.2.0.tar.gz
cd VTK-8.2.0
mkdir build
cd build
cmake -DCMAKE_INSTALL_PREFIX=$VTK_HOME -DVTK_WRAP_JAVA=TRUE ..
make install
</pre></div>
</div>
<p>Note that <code class="docutils literal notranslate"><span class="pre">$VTK_HOME</span></code> must be defined prior to invocation of cmake. The VTK Java bindings jar-file <code class="docutils literal notranslate"><span class="pre">vtk.jar</span></code> and a multitude of dynamically linked sub-libraries are installed into <code class="docutils literal notranslate"><span class="pre">$VTK_HOME/lib</span></code>.</p>
</section>
<section id="mowgli">
<h3>mowgli<a class="headerlink" href="#mowgli" title="Permalink to this headline"></a></h3>
<p>Implementation of surfaces, distribution samplers, and other low-level building blocks are currently part of the <strong>mowgli</strong> library. Clone it from &lt;<a class="reference external" href="http://hardin.jhuapl.edu:8080/scm/zimmemi1/mowgli">http://hardin.jhuapl.edu:8080/scm/zimmemi1/mowgli</a>&gt;. Further documentation can be found <em>here</em>.</p>
</section>
<section id="lestat">
<h3>lestat<a class="headerlink" href="#lestat" title="Permalink to this headline"></a></h3>
<p>TODO: copy from lestat getting-started docs</p>
</section>
</section>
<section id="installation-and-test-run">
<h2>Installation and test run<a class="headerlink" href="#installation-and-test-run" title="Permalink to this headline"></a></h2>
<p>Get the code from &lt;<a class="reference external" href="http://hardin.jhuapl.edu:8080/scm/zimmemi1/dragion">http://hardin.jhuapl.edu:8080/scm/zimmemi1/dragion</a>&gt;. Assume that it is cloned into <cite>~/dragion</cite>.</p>
<p>For now the classpath and runtime library path are passed explicitly to the JRE from the command line (this will probably chang&gt;</p>
<p>To print a helpful message, try:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">DYLD_LIBRARY_PATH</span><span class="o">=~/</span><span class="n">local</span><span class="o">/</span><span class="n">vtk</span><span class="o">-</span><span class="mf">8.2</span><span class="o">/</span><span class="n">lib</span> \
<span class="n">java</span> <span class="o">-</span><span class="n">Djava</span><span class="o">.</span><span class="n">library</span><span class="o">.</span><span class="n">path</span><span class="o">=/</span><span class="n">Users</span><span class="o">/</span><span class="n">zimmemi1</span><span class="o">/</span><span class="n">local</span><span class="o">/</span><span class="n">vtk</span><span class="o">-</span><span class="mf">8.2</span><span class="o">/</span><span class="n">lib</span> \
<span class="o">-</span><span class="n">classpath</span> <span class="o">~/</span><span class="n">dragion</span><span class="o">/</span><span class="nb">bin</span><span class="p">:</span><span class="o">~/</span><span class="n">dragion</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">commons</span><span class="o">-</span><span class="n">math3</span><span class="o">-</span><span class="mf">3.6.1</span><span class="o">/</span><span class="n">commons</span><span class="o">-</span><span class="n">math3</span><span class="o">-</span><span class="mf">3.6.1</span><span class="o">.</span><span class="n">jar</span><span class="p">:</span><span class="o">~/</span><span class="n">dragion</span><span class="o">/</span><span class="n">lib</span><span class="o">/</span><span class="n">guava</span><span class="o">-</span><span class="mf">28.1</span><span class="o">/</span><span class="n">guava</span><span class="o">-</span><span class="mf">28.1</span><span class="o">-&gt;</span>
<span class="n">dragion</span><span class="o">.</span><span class="n">Main</span>
</pre></div>
</div>
</section>
</section>
</div>
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</ul>
</li>
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<li>Previous: <a href="index.html" title="previous chapter">dragion</a></li>
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<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
<title>dragion &#8212; dragion documentation</title>
<link rel="stylesheet" type="text/css" href="_static/pygments.css" />
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<section id="dragion">
<h1>dragion<a class="headerlink" href="#dragion" title="Permalink to this headline"></a></h1>
<p>A Java API for modeling electrostatic interactions between atmospheric ions and charged surfaces, including:</p>
<ul class="simple">
<li><p>arbitrarily shaped homogeneous dielectrics and (biased or floating) conductors</p></li>
<li><p>self-consistent surface charging through ion collection</p></li>
<li><p>ion diffusion and mobility (in the presence of external and surface fields)</p></li>
<li><p>volumetric ion sources and collisional loss process</p></li>
</ul>
<div class="toctree-wrapper compound">
<p class="caption" role="heading"><span class="caption-text">Contents:</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="getstarted.html">Getting started</a><ul>
<li class="toctree-l2"><a class="reference internal" href="getstarted.html#dependencies">Dependencies</a></li>
<li class="toctree-l2"><a class="reference internal" href="getstarted.html#installation-and-test-run">Installation and test run</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="physvalid.html">Physics validation</a><ul>
<li class="toctree-l2"><a class="reference internal" href="examples/unipolar_source_loss.html">Unipolar ions: uniform source and loss</a></li>
<li class="toctree-l2"><a class="reference internal" href="examples/bipolar_source_loss.html">Bipolar ions: uniform source and loss (unfinished)</a></li>
<li class="toctree-l2"><a class="reference internal" href="examples/relaxation_sphere.html">Unipolar relaxation</a></li>
</ul>
</li>
</ul>
</div>
</section>
<section id="indices-and-tables">
<h1>Indices and tables<a class="headerlink" href="#indices-and-tables" title="Permalink to this headline"></a></h1>
<ul class="simple">
<li><p><a class="reference internal" href="genindex.html"><span class="std std-ref">Index</span></a></p></li>
<li><p><a class="reference internal" href="py-modindex.html"><span class="std std-ref">Module Index</span></a></p></li>
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<h3>Navigation</h3>
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<ul>
<li class="toctree-l1"><a class="reference internal" href="getstarted.html">Getting started</a></li>
<li class="toctree-l1"><a class="reference internal" href="physvalid.html">Physics validation</a></li>
</ul>
<div class="relations">
<h3>Related Topics</h3>
<ul>
<li><a href="#">Documentation overview</a><ul>
<li>Next: <a href="getstarted.html" title="next chapter">Getting started</a></li>
</ul></li>
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<h3 id="searchlabel">Quick search</h3>
<div class="searchformwrapper">
<form class="search" action="search.html" method="get">
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<input type="submit" value="Go" />
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<script>$('#searchbox').show(0);</script>
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Powered by <a href="http://sphinx-doc.org/">Sphinx 4.5.0</a>
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
<title>Physics validation &#8212; dragion documentation</title>
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<h1>Physics validation<a class="headerlink" href="#physics-validation" title="Permalink to this headline"></a></h1>
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<li class="toctree-l1"><a class="reference internal" href="examples/unipolar_source_loss.html">Unipolar ions: uniform source and loss</a></li>
<li class="toctree-l1"><a class="reference internal" href="examples/bipolar_source_loss.html">Bipolar ions: uniform source and loss (unfinished)</a></li>
<li class="toctree-l1"><a class="reference internal" href="examples/relaxation_sphere.html">Unipolar relaxation</a></li>
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<li class="toctree-l1 current"><a class="current reference internal" href="#">Physics validation</a><ul>
<li class="toctree-l2"><a class="reference internal" href="examples/unipolar_source_loss.html">Unipolar ions: uniform source and loss</a></li>
<li class="toctree-l2"><a class="reference internal" href="examples/bipolar_source_loss.html">Bipolar ions: uniform source and loss (unfinished)</a></li>
<li class="toctree-l2"><a class="reference internal" href="examples/relaxation_sphere.html">Unipolar relaxation</a></li>
</ul>
</li>
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<h3>Related Topics</h3>
<ul>
<li><a href="index.html">Documentation overview</a><ul>
<li>Previous: <a href="getstarted.html" title="previous chapter">Getting started</a></li>
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.. dragion documentation master file, created by
sphinx-quickstart on Thu Aug 25 16:19:38 2022.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
dragion
=======
A Java API for modeling electrostatic interactions between atmospheric ions and charged surfaces, including:
- arbitrarily shaped homogeneous dielectrics
- biased and floating conductors
- self-consistent surface charging through ion collection
- ion diffusion
- ion mobility (in the presence of external and surface fields)
- volumetric ion sources
- collisional ion loss
----
Getting started
===============
Get the code from <http://hardin.jhuapl.edu:8080/scm/zimmemi1/dragion>. Assume that it is cloned into `~/dragion`.
You will need to compile [VTK 8.2](vtk.org/download) from source, with Java bindings enabled. Typically on MacOS or Linux, using [CMake](https://cmake.org/), this looks like::
wget https://www.vtk.org/files/release/8.2/VTK-8.2.0.tar.gz
tar xf VTK-8.2.0.tar.gz
cd VTK-8.2.0
mkdir build
cd build
cmake -DCMAKE_INSTALL_PREFIX=~/local/vtk-8.2 -DVTK_WRAP_JAVA=TRUE ..
make install
This installs vtk.jar and a multitude of dynamically linked VTK sub-libraries into `~/local/vtk-8.2/lib`.
For now the classpath and runtime library path are passed explicitly to the JRE from the command line (this will probably change in a future release, i.e. by packing all required libs and natives into a single runnable jar).
To print a helpful message, try::
DYLD_LIBRARY_PATH=~/local/vtk-8.2/lib \
java -Djava.library.path=/Users/zimmemi1/local/vtk-8.2/lib \
-classpath ~/dragion/bin:~/dragion/lib/commons-math3-3.6.1/commons-math3-3.6.1.jar:~/dragion/lib/guava-28.1/guava-28.1-jre.jar:~/local/vtk-8.2/lib/vtk.jar:~/dragion/lib/tomlj-1.0.0/tomlj-1.0.0.jar:~/dragion/lib/commons-cli-1.5.0/commons-cli-1.5.0.jar:~/dragion/lib/antlr4-runtime-4.7.2/antlr4-runtime-4.7.2.jar \
dragion.Main
.. toctree::
:maxdepth: 2
:caption: Contents:
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`

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@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

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# Configuration file for the Sphinx documentation builder.
#
# This file only contains a selection of the most common options. For a full
# list see the documentation:
# https://www.sphinx-doc.org/en/master/usage/configuration.html
# -- Path setup --------------------------------------------------------------
# If extensions (or modules to document with autodoc) are in another directory,
# add these directories to sys.path here. If the directory is relative to the
# documentation root, use os.path.abspath to make it absolute, like shown here.
#
# import os
# import sys
# sys.path.insert(0, os.path.abspath('.'))
# -- Project information -----------------------------------------------------
project = 'dragion'
copyright = '2022, zimmemi1'
author = 'zimmemi1'
# -- General configuration ---------------------------------------------------
# Add any Sphinx extension module names here, as strings. They can be
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
extensions = [
]
# Add any paths that contain templates here, relative to this directory.
templates_path = ['_templates']
# List of patterns, relative to source directory, that match files and
# directories to ignore when looking for source files.
# This pattern also affects html_static_path and html_extra_path.
exclude_patterns = []
# -- Options for HTML output -------------------------------------------------
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
#
html_theme = 'alabaster'
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
html_static_path = ['_static']

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Bipolar ions: uniform source and loss (unfinished)
==================================================
Problem Statement
-----------------
The bipolar balance equation (in the absence of aerosol particles) is
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}
where :math:`n_\pm` are respective ion concentrations and :math:`\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}` [Harrison and Tammet]. Sans aerosols, :math:`n_+=n_-`.
**dragion** can be tested against this bipolar balance equation with uniform ``VolumeSource`` and ``CollisionalSinkBipolar`` instances for each ion species.
The implementation is forthcoming, to be based on ``dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java``.
The ``VolumeSource`` objects uniformly distribute positive and negative ions across the simulation ``Box`` at a steady rate, representing the constant source term :math:`q`.
The ``CollisionalSinkBipolar`` algorithm simulates the respective loss terms :math:`- \alpha n_+n_-` by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing :math:`n_+n_-` in each grid cell
(2) removing :math:`\alpha \: n_+n_- \: \mathrm{d}t \: \mathrm{d}^3r \: / np2c` macroparticles of each species from the grid cell (leaving zero if exceeding existing number)
We would expect the simulated :math:`\frac{\mathrm{d}n}{\mathrm{d}t}` to equal :math:`0`, integrated over many timesteps; in other words, the simulated concentration should be constant and the positive and negative concentrations equal.
Numerical Validation
--------------------
Taking :math:`q = 4\times10^6 \mathrm{m}^{-3} \mathrm{s}^{-1}` and :math:`\frac{\mathrm{d}n_+}{\mathrm{d}t}=0` yields a steady-state concentration of :math:`n_{equil}=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG].
TODO

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Unipolar relaxation
===================
Problem Statement
-----------------
Consider an initially uncharged conducting sphere placed in a uniform, cold, unipolar ion bath.
After a while we bias the sphere to a polarity opposite that of the ions, and then immediately disconnect it, leaving the sphere floating at an attractive nonzero potential.
The resulting unipolar current can be derived from the electric field
.. math::
:nowrap:
\begin{eqnarray}
E_r=\frac{Q}{4 \pi \varepsilon_0 r^2}
\end{eqnarray}
and the definition of mobility
.. math::
:nowrap:
\begin{eqnarray}
\vec{v} = \mu \vec{E}
\end{eqnarray}
to get
.. math::
:nowrap:
\begin{eqnarray}
I = n q v_r A = n q \cdot \mu E_r \cdot 4 \pi R^2
\end{eqnarray}
where :math:`n` is ion concentration, :math:`q` is ion charge, and :math:`R` is radius of the sphere. This yields the differential equation for charge on the sphere
.. math::
:nowrap:
\begin{eqnarray}
\dot{Q}=n q \mu \cdot \frac{Q}{4 \pi \varepsilon_0 R^2} \cdot 4 \pi R^2=Q \frac{n q \mu}{\varepsilon_0}
\end{eqnarray}
Converting :math:`Q` to :math:`V` using the definition of capacitance
.. math::
:nowrap:
\begin{eqnarray}
C V = Q
\end{eqnarray}
and noting that
.. math::
:nowrap:
\begin{eqnarray}
\sigma = n |q| \mu
\end{eqnarray}
the solution is
.. math::
:nowrap:
\begin{eqnarray}
V(t) = V_0 \exp\left(-\frac{t}{\varepsilon_0/\sigma}\right)
\end{eqnarray}
The negative sign in the exponential comes from the sign of :math:`q` relative to :math:`V_0`.
Numerical Validation
--------------------
Runs simulating the relaxation phenomenon for a conducting sphere can be found in ``dragion/runs/sphere``.
Initially, positive unipolar ions are loaded uniformly into the domain, with concentration :math:`n=\sqrt{\frac{q}{\alpha}}=2\times10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG], and a sphere of radius :math:`R=0.5 \mathrm{m}` is biased to :math:`-10 \mathrm{V}` then left floating.
Note: this simulation does not include a uniform background source or collisional sink.
The resulting time-progression is illustrated by a sequence of 3D screenshots:
- :math:`t=0 \mathrm{s}`
.. image:: relaxation-sphere-3d_0.png
- :math:`t=500 \mathrm{s}`
.. image:: relaxation-sphere-3d_1.png
- :math:`t=1000 \mathrm{s}`
.. image:: relaxation-sphere-3d_2.png
As the sphere collects nearby ions its (negative) charge level and electric field strength diminish.
The voltage drop is exponential, and its slope (in log-voltage) can be used to back out an estimate of atmospheric conductivity, :math:`\sigma = \mathrm{slope} \cdot \varepsilon_0`.
Here is a plot of (log10) voltage over time:
.. image:: relaxation-sphere.png
Notice that overall the **dragion** sphere relaxes more slowly than predicted by theory.
However, there is a slight inflection in the simulated voltage at :math:`t=1200 \mathrm{s}`.
A comparison of slopes before and after this knee, and the full end-to-end slope provide estimates for conductivity, using :math:`\sigma = \mathrm{slope} \cdot \varepsilon_0`:
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
| Fit | Slope (relaxation rate) :math:`[\mathrm{s}^{-1}]` | :math:`\sigma [\mathrm{S}/\mathrm{m}]` |
+=====================================+====================================================+==================================================+
| End-to-end | -0.0011 | :math:`0.95\times10^{-14}` |
| | | |
| Before :math:`t=1200 \: \mathrm{s}` | -0.0009 | :math:`0.79\times10^{-14}` |
| | | |
| After :math:`t=1200 \:\mathrm{s}` | -0.0013 | :math:`1.2\times10^{-14}` |
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
| Theoretical | -0.0016 | :math:`1.4\times10^{-14}` |
+-------------------------------------+----------------------------------------------------+--------------------------------------------------+
*More discussion is needed. Why is there an inflection point? It is probably related to domain shape and size. Why is the late-time estimate the best?*
At very late times the near-surface ion concentration will be depleted, so without a background source charging will eventually halt. In this regime the assumption of steady near-surface concentration would be violated and the theoretical model breaks down.

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Unipolar ions: uniform source and loss
======================================
Problem Statement
-----------------
The bipolar balance equation (in the absence of aerosol particles) is
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n_+}{\mathrm{d}t} = q - \alpha n_+ n_- \\
\frac{\mathrm{d}n_-}{\mathrm{d}t} = q - \alpha n_+ n_-
\end{eqnarray}
where :math:`n_\pm` are respective ion concentrations and :math:`\alpha\sim10^{-12} \: \mathrm{m}^3 \: \mathrm{s}^{-1}` [Harrison and Tammet]. Sans aerosols, :math:`n_+=n_-`, so that a single species reads
.. math::
:nowrap:
\begin{eqnarray}
\frac{\mathrm{d}n}{\mathrm{d}t} = q - \alpha n^2
\end{eqnarray}
**dragion** can be tested against this unipolar balance equation with a uniform ``VolumeSource`` and ``CollisionalSinkUnipolar``; the implementation is in ``dragion/simulations/unipolar/sourcesinktest/SourceSinkTest.java``.
The ``VolumeSource`` uniformly distributes particles across the simulation ``Box`` at a steady rate, representing the constant source term :math:`q`.
The ``CollisionalSinkUnipolar`` algorithm simulates the loss term :math:`- \alpha n^2` by overlaying a uniform grid on the simulation domain, and at each timestep
(1) computing :math:`n^2` in each grid cell
(2) removing :math:`n^2 \: \alpha \: \mathrm{d}t \: \mathrm{d}^3r \: / \: np2c` macroparticles per grid cell (leaving zero if exceeding the existing number)
We would expect the simulated :math:`\frac{\mathrm{d}n}{\mathrm{d}t}` to equal :math:`0`, integrated over many timesteps; in other words, the simulated concentration should be constant, with ion production in numerical balance with loss.
Numerical Validation
--------------------
Taking :math:`q = 4 \times 10^6 \: \mathrm{m}^{-3} \: \mathrm{s}^{-1}` and :math:`\mathrm{d}n/\mathrm{d}t = 0` yields a theoretical steady-state unipolar concentration of :math:`n_{equil}=\sqrt{q / \alpha = 2 \times 10^9 \: \mathrm{m}^{-3}` [Lorenz, MMRTG].
The ``SourceSinkTest`` starts with an empty 5 m simulation box. The ``VolumeSource`` fills the domain with macroparticles, until the ion concentration reaches :math:`n_{equil}` and the ``CollisionalSinkUnipolar`` removes as many ions as are added, on average.
Crucial simulation parameters are the macroparticle "size", controlled by :math:`np2c`, and the grid spacing used for particle-particle collisions. For this numerical experiment the collision cell-size is the domain length divided by 64.
At :math:`np2c=10^7` the mean concentration peaks rapidly but proceeds to fall off unphysically: the number of macroparticles per grid-cell is far too low, with cell size / particle size ratio :math:=`(5 \mathrm{m}/64)^3/(n_{equil}/np2c) = 0.1`.
Reducing :math:`np2c` by two orders of magnitude increases the # particles per grid cell to :math:`\sim 10`, and the expected steady-state concentration is recovered, with small oscillations about the mean.
.. image:: coll-equil.png

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Getting started
===============
Dependencies
------------
VTK
^^^^^^^
You will need to compile [VTK 8.2](vtk.org/download) from source, with Java bindings enabled. Typically on MacOS or Linux, using [CMake](https://cmake.org/), this looks like::
wget https://www.vtk.org/files/release/8.2/VTK-8.2.0.tar.gz
tar xf VTK-8.2.0.tar.gz
cd VTK-8.2.0
mkdir build
cd build
cmake -DCMAKE_INSTALL_PREFIX=$VTK_HOME -DVTK_WRAP_JAVA=TRUE ..
make install
Note that ``$VTK_HOME`` must be defined prior to invocation of cmake. The VTK Java bindings jar-file ``vtk.jar`` and a multitude of dynamically linked sub-libraries are installed into ``$VTK_HOME/lib``. This location will be part of the classpath when running dragion.
mowgli
^^^^^^
Implementation of surfaces, distribution samplers, and other low-level building blocks are currently part of the **mowgli** library. Clone it from <http://hardin.jhuapl.edu:8080/scm/zimmemi1/mowgli>. Further documentation can be found *here*.
lestat
^^^^^^
TODO: copy from lestat getting-started docs
Installation and test run
-------------------------
Get the code from <http://hardin.jhuapl.edu:8080/scm/zimmemi1/dragion>. Assume that it is cloned into `~/dragion`.
For now the classpath and runtime library path are passed explicitly to the JRE from the command line (this will probably chang>
To print a helpful message, try::
DYLD_LIBRARY_PATH=~/local/vtk-8.2/lib \
java -Djava.library.path=/Users/zimmemi1/local/vtk-8.2/lib \
-classpath ~/dragion/bin:~/dragion/lib/commons-math3-3.6.1/commons-math3-3.6.1.jar:~/dragion/lib/guava-28.1/guava-28.1->
dragion.Main

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.. dragion documentation master file, created by
sphinx-quickstart on Thu Aug 25 16:19:38 2022.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
dragion
=======
A Java API for modeling electrostatic interactions between atmospheric ions and charged surfaces, including:
- arbitrarily shaped homogeneous dielectrics and (biased or floating) conductors
- self-consistent surface charging through ion collection
- ion diffusion and mobility (in the presence of external and surface fields)
- volumetric ion sources and collisional loss process
.. toctree::
:maxdepth: 2
:caption: Contents:
getstarted.rst
physvalid.rst
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`

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Physics validation
==================
.. toctree::
:maxdepth: 1
examples/unipolar_source_loss.rst
examples/bipolar_source_loss.rst
examples/relaxation_sphere.rst

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<TeXmacs|1.99.12>
<style|generic>
<\body>
<paragraph|Isolated sphere>
For a uniformly charged (and electrically isolated) sphere,
<\equation*>
E<rsub|r>=<frac|Q|4*\<pi\>*\<varepsilon\><rsub|0> r<rsup|2>>
</equation*>
where initially <math|Q=Q<rsub|0>>.
<\equation*>
<wide|v|\<vect\>>=\<mu\>*<wide|E|\<vect\>>
</equation*>
<\equation*>
I=-n*q*<wide|v|\<vect\>><rsub|r>\<cdot\><wide|A|\<vect\>>=n*q*\<mu\>*E<rsub|r>\<cdot\>4*\<pi\>*R<rsup|2>
</equation*>
Note that <math|q=Z e> must have opposite sign of <math|E<rsub|r>> (and
<math|Q<rsub|0>>) to be attracted.
<\equation*>
<wide|Q|\<dot\>>=n*q*\<mu\>*<frac|Q|4*\<pi\>*\<varepsilon\><rsub|0>
R<rsup|2>>\<cdot\>4*\<pi\>*R<rsup|2>
</equation*>
If <math|n> is steady and uniform then
<\equation*>
Q=Q<rsub|0> exp<around*|(|-<frac|n <around*|\||q|\|>
\<mu\>|\<varepsilon\><rsub|0>>*t|)>
</equation*>
Note that <math|\<sigma\>\<equiv\>n <around*|\||q|\|> \<mu\>> so that
<\equation*>
Q=Q<rsub|0> exp<around*|(|-<frac|\<sigma\>|\<varepsilon\><rsub|0>>*t|)>
</equation*>
\;
<paragraph|With constant windspeed>
The timestep equation is
<\equation*>
v<around*|(|t|)>=v<around*|(|0|)>*exp<around*|(|-<frac|\<mathd\>t|\<tau\><rsub|c>>|)>+<around*|(|\<mu\>*E+u|)><around*|(|1-exp<around*|(|-<frac|\<mathd\>t|\<tau\><rsub|c>>|)>|)>
</equation*>
and with <math|\<mathd\>t\<gg\>\<tau\><rsub|c>>,
<\equation*>
v<around*|(|t|)>=\<mu\>*E+u
</equation*>
Total current (positive = to the ball) is
<\equation*>
I=-<big|int>n*q*<wide|v|\<vect\>>\<cdot\>\<mathd\><wide|A|\<vect\>>=-<big|int>n*q*<around*|(|\<mu\>*<wide|E|\<vect\>>+<wide|u|\<vect\>>|)>\<cdot\>\<mathd\><wide|A|\<vect\>>
</equation*>
The electric field/mobility part is
<\equation*>
I<rsub|\<mu\>>=-<big|int>n*q*<around*|(|\<mu\>*<wide|E|\<vect\>>|)>\<cdot\>\<mathd\><wide|A|\<vect\>>=-n*q*<around*|(|4*\<pi\>*R<rsup|2>
\<mu\>*E<rsub|r>|)>=-n*q\<cdot\>4*\<pi\>*R<rsup|2>
\<mu\>\<cdot\><frac|Q|4*\<pi\>*\<varepsilon\><rsub|0>*R<rsup|2>>=-<frac|n*q*\<mu\>|\<varepsilon\><rsub|0>>*Q=-<frac|\<sigma\>|\<varepsilon\><rsub|0>>*Q
</equation*>
and the wind contribution is
<\equation*>
I<rsub|u>=-<big|int>n*q*<around*|(|<wide|u|\<vect\>>|)>\<cdot\>\<mathd\><wide|A|\<vect\>>=-n*q*v<rsub|wind><below|<big|int>|hemi>*<wide|x|^>\<cdot\>\<mathd\><wide|A|\<vect\>>*
</equation*>
<\equation*>
<below|<big|int>|hemi>*<wide|x|^>\<cdot\>\<mathd\><wide|A|\<vect\>>*=R<rsup|2><below|<above|<big|int>|\<pi\>/2>|0>\<mathd\><around*|(|cos
\<theta\>|)><below|<above|<big|int>|2*\<pi\>>|0>\<mathd\>\<phi\>
<wide|x|^>\<cdot\><wide|r|^>=2*\<pi\>*R<rsup|2><below|<above|<big|int>|\<pi\>/2>|0>\<mathd\><around*|(|cos
\<theta\>|)> cos \<theta\>=2*\<pi\>*R<rsup|2><around*|{|<frac|1|2>*cos<rsup|2>
<frac|\<pi\>|2>-<frac|1|2>*cos<rsup|2> 0|}>=-\<pi\>*R<rsup|2>
</equation*>
<\equation*>
I<rsub|u>=n*q*v<rsub|wind>*\<pi\>*R<rsup|2>
</equation*>
so the total current becomes
<\equation*>
I=-<frac|\<sigma\>|\<varepsilon\><rsub|0>>*Q+n*q
v<rsub|wind>*\<pi\>*R<rsup|2>
</equation*>
where
<\equation*>
\<tau\><rsub|\<sigma\>>=<frac|\<varepsilon\><rsub|0>|\<sigma\>>
</equation*>
Grouping terms,
<\equation*>
<wide|Q|\<dot\>>=-<frac|Q|\<tau\><rsub|\<sigma\>>>+I<rsub|wind>
</equation*>
<\equation*>
Q=C \<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*\<tau\><rsub|\<sigma\>>
</equation*>
here\
<\equation*>
Q<rsub|0>=C +I<rsub|wind>*\<tau\><rsub|\<sigma\>><space|1em>\<longrightarrow\>\<space\>\<space\><application-space|1em>C=Q<rsub|0>-I<rsub|wind>*\<tau\><rsub|\<sigma\>>
</equation*>
<\equation*>
Q=Q<rsub|0> \<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|)>
</equation*>
<paragraph|Wake/collection-area correction>
<\with|color|pastel brown>
There is an extra factor <math|w> on the <math|Q>-term when the windspeed
is great (because of the wake),
<\equation*>
<wide|Q|\<dot\>>=-<frac|1|w>*<frac|Q|\<tau\><rsub|\<sigma\>>>+I<rsub|wind>
</equation*>
<\equation*>
Q=C \<bbb-e\><rsup|-t/w*\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*w**\<tau\><rsub|\<sigma\>>
</equation*>
<\equation*>
Q<rsub|0>=C +I<rsub|wind>*w**\<tau\><rsub|\<sigma\>><space|1em>\<longrightarrow\>\<space\>\<space\><application-space|1em>C=Q<rsub|0>-I<rsub|wind>*w**\<tau\><rsub|\<sigma\>>
</equation*>
<\equation*>
Q=Q<rsub|0> \<bbb-e\><rsup|-t/w*\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*w**\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/w*\<tau\><rsub|\<sigma\>>>|)>
</equation*>
</with>
The quasisteady velocity field is
<\equation*>
<wide|v|\<vect\>>=\<mu\>*<wide|E|\<vect\>>+<wide|u|\<vect\>>=\<mu\>*E*<wide|r|^>+<around*|\||u|\|>*<wide|u|^>
</equation*>
<\equation*>
<wide|v|\<vect\>>\<cdot\><wide|r|^>=\<mu\>*E*<around*|(|<wide|r|^>\<cdot\><wide|r|^>|)>+<around*|\||u|\|>*<around*|(|<wide|u|^>\<cdot\><wide|r|^>|)>=\<mu\>*E-<around*|\||u|\|>*cos
\<eta\>
</equation*>
where <math|cos \<eta\>\<equiv\>-<wide|u|^>\<cdot\><wide|r|^>>. The only
particles at <math|r=R> that can reach the ball have
<math|<wide|v|\<vect\>>\<cdot\><wide|r|^>\<less\>0>,
<\equation*>
\<mu\>*E-<around*|\||u|\|>*cos \<eta\>\<less\>0
</equation*>
<\equation*>
cos \<eta\>\<less\><frac|\<mu\>*E|<around*|\||u|\|>>
</equation*>
Total current to the ball is
<\equation*>
I=-<below|<big|int>|<wide|v|\<vect\>>\<cdot\>\<mathd\><wide|A|\<vect\>>\<less\>0>n*q*<wide|v|\<vect\>>\<cdot\>\<mathd\><wide|A|\<vect\>>=+2*\<pi\><below|<above|<big|int>|cos
\<eta\><rsub|max>>|0>n*q*<around*|(|\<mu\>*E-<around*|\||u|\|>*cos
\<eta\>|)>R<rsup|2>*\<mathd\> <around*|(|cos \<eta\>|)>
</equation*>
where
<\equation*>
cos \<eta\><rsub|max>=<frac|\<mu\>*E|<around*|\||u|\|>><space|1em>limited
to -1\<less\>cos \<eta\><rsub|max>\<less\>1
</equation*>
With <math|<around*|\||u|\|>=0> the entire sphere has the possibility of
collecting particles; <math|\<eta\><rsub|max>=\<pi\>>.
With <math|\<mu\>*E=0> (a grounded sphere), or in the high-wind limit
<math|<around*|\||u|\|>\<gg\>\<mu\>*E>, only half the sphere (the windward
side) is collecting; <math|\<eta\><rsub|max>=<frac|\<pi\>|2>>.
Assuming positive carriers, an attracting sphere has <math|E\<less\>0>;
<math|\<eta\><rsub|max>\<gtr\>\<pi\>/2> and some of the back side is
shielded. The wind will eventually force a sign flip to <math|E\<gtr\>0>,
so that <math|\<eta\><rsub|max>\<less\>\<pi\>/2> and some of the front side
is shielded.
The mobility current is
<\equation*>
I<rsub|mob>=2*\<pi\>*n*q*\<mu\>*E*R<rsup|2><below|<above|<big|int>|\<eta\><rsub|max>>|0>*\<mathd\>
<around*|(|cos \<eta\>|)>=2*\<pi\>*n*q*\<mu\>*E*R<rsup|2>*<around*|(|cos
\<eta\><rsub|max>-1|)>
</equation*>
and the wind current is
<\equation*>
I<rsub|wind>=-2*\<pi\>*n*q*<around*|\||u|\|>*R<rsup|2>*<below|<above|<big|int>|\<eta\><rsub|max>>|0><around*|(|cos
\<eta\>|)>*\<mathd\> <around*|(|cos \<eta\>|)>=-2*\<pi\>*n*q*<around*|\||u|\|>*R<rsup|2><around*|(|<frac|cos<rsup|2>
\<eta\><rsub|max>|2>-<frac|1|2>|)>
</equation*>
so the total current is
<\equation*>
I=I<rsub|mob>+I<rsub|wind>=2*\<pi\>*n*q*R<rsup|2>*<around*|[|\<mu\>*E*<around*|(|cos
\<eta\><rsub|max>-1|)>-<frac|<around*|\||u|\|>|2>*<around*|(|cos<rsup|2>
\<eta\><rsub|max>-1|)>|]>
</equation*>
<\equation*>
<frac|\<mathd\>Q|\<mathd\>t>=2*\<pi\>*n*q*R<rsup|2>*<around*|[|\<mu\>*E*<around*|(|cos
\<eta\><rsub|max>-1|)>-<frac|<around*|\||u|\|>|2>*<around*|(|cos<rsup|2>
\<eta\><rsub|max>-1|)>|]>
</equation*>
<paragraph|Solving>
By the relation between sphere charge and surface electric field,
<\equation*>
<frac|\<mathd\>Q|\<mathd\>t>=4*\<pi\>*\<varepsilon\><rsub|0>*R<rsup|2>*<frac|\<mathd\>E|\<mathd\>t>
</equation*>
<\equation*>
<frac|\<mathd\>E|\<mathd\>t>=<frac|n*q|2*\<varepsilon\><rsub|0>>*<around*|[|\<mu\>*E*<around*|(|cos
\<eta\><rsub|max>-1|)>-<frac|<around*|\||u|\|>|2>*<around*|(|cos<rsup|2>
\<eta\><rsub|max>-1|)>|]>
</equation*>
<\with|color|pastel grey>
Without the clamps on <math|cos \<eta\><rsub|max>>,
<\equation*>
<frac|\<mathd\>E|\<mathd\>t>=<frac|n*q|2*\<varepsilon\><rsub|0>>*<around*|[|\<mu\>*E*<around*|(|<frac|\<mu\>*E|<around*|\||u|\|>>-1|)>-<frac|<around*|\||u|\|>|2>*<around*|(|<frac|\<mu\>*E|<around*|\||u|\|>>-1|)>|]>
</equation*>
<\equation*>
<frac|\<mathd\>E|\<mathd\>t>=A*E<rsup|2>+B*E+C
</equation*>
<\equation*>
A=C*<frac|\<mu\><rsup|2>|<around*|\||u|\|>>
</equation*>
<\equation*>
B=<frac|n*q|2*\<varepsilon\><rsub|0>>*<around*|(|-\<mu\>-<frac|\<mu\>|2>|)>=-<frac|3|2>*C*\<mu\>
</equation*>
<\equation*>
C=<frac|n*q|2*\<varepsilon\><rsub|0>>
</equation*>
</with>
\;
<paragraph|Regimes>
<\equation*>
Q<rsub|0>=C*\<Phi\><rsub|0>=4*\<pi\>*\<varepsilon\><rsub|0>*R*\<Phi\><rsub|0>
</equation*>
<\equation*>
\<tau\><rsub|\<sigma\>>=<frac|\<varepsilon\><rsub|0>|\<sigma\>>
</equation*>
<\equation*>
I<rsub|wind>=n*Z*q<rsub|e>*v<rsub|wind> \<pi\>*R<rsup|2>
</equation*>
The only dimensionless ratio is
<\equation*>
\<frak-I\>\<equiv\><frac|I<rsub|wind>*\<tau\><rsub|\<sigma\>>|Q<rsub|0>>
</equation*>
A traditional relaxation probe has <math|\<frak-I\>=0> and pure wind has
<math|\<frak-I\>\<rightarrow\>\<infty\>>.
The ratio of charges is
<\equation*>
\<frak-I\>=<frac|n*Z*q<rsub|e>*v<rsub|wind> \<pi\>*R<rsup|2>
<frac|\<varepsilon\><rsub|0>|\<sigma\>>|4*\<pi\>*\<varepsilon\><rsub|0>*R*\<Phi\><rsub|0>>=<frac|<frac|\<sigma\>|\<mu\>>*v<rsub|wind>*R|\<sigma\>*4*\<Phi\><rsub|0>>\<propto\><frac|v<rsub|wind>*R|\<mu\>
\<Phi\><rsub|0>>=<frac|v<rsub|wind>|\<mu\>*E<rsub|0>>
</equation*>
and checking units,
<\equation*>
<around*|[|<frac|v<rsub|wind>*R|\<mu\>
\<Phi\><rsub|0>>|]>=<around*|[|<frac|v<rsub|wind>|\<mu\>*E<rsub|0>>|]>=1
</equation*>
\;
<paragraph|Conductivity from slope (no area correction)>
<\equation*>
Q=Q<rsub|0> \<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|)>
</equation*>
\;
At early-enough times
\;
<\equation*>
Q=Q<rsub|0> <around*|(|1-<frac|t|\<tau\><rsub|\<sigma\>>>|)>+I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-<around*|(|1-<frac|t|\<tau\><rsub|\<sigma\>>>|)>|)>
</equation*>
<\equation*>
Q=Q<rsub|0> <around*|(|1-<frac|t|\<tau\><rsub|\<sigma\>>>|)>+I<rsub|wind>*t
</equation*>
then
<\equation*>
slope=<frac|\<mathd\>|\<mathd\>t> log
<around*|(|Q|)>=<frac|\<mathd\>|\<mathd\>t> log <around*|(|Q<rsub|0>
<around*|(|1-<frac|t|\<tau\><rsub|\<sigma\>>>|)>+I<rsub|wind>*t|)>=<frac|\<mathd\>|\<mathd\>t>
log <around*|(|Q<rsub|0> +<around*|(|I<rsub|wind>-<frac|Q<rsub|0>|\<tau\><rsub|\<sigma\>>>|)>*t|)>
</equation*>
<\equation*>
slope=<frac|<around*|(|I<rsub|wind>-<frac|Q<rsub|0>|\<tau\><rsub|\<sigma\>>>|)>|Q<rsub|0>+<around*|(|I<rsub|wind>-<frac|Q<rsub|0>|\<tau\><rsub|\<sigma\>>>|)>*t>
</equation*>
and at <math|t=0> this becomes
<\equation*>
slope=<frac|I<rsub|wind>|Q<rsub|0>>-<frac|\<sigma\>|\<varepsilon\><rsub|0>>
</equation*>
\;
\;
At late-enough times
<\equation*>
slope=<frac|\<mathd\>|\<mathd\>t> log
<around*|(|I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|)>|)>=<frac|1|I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|)>>*<frac|\<mathd\>|\<mathd\>t>
I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|)>
</equation*>
<\equation*>
=<frac|1|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>>*<around*|(|+<frac|\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|\<tau\><rsub|\<sigma\>>>|)>=<frac|\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>|1-\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>>*<frac|\<sigma\>|\<varepsilon\><rsub|0>>\<approx\><frac|\<sigma\>|\<varepsilon\><rsub|0>>*\<bbb-e\><rsup|-t/\<tau\><rsub|\<sigma\>>>
</equation*>
<paragraph|Saturation potential>
The saturation potential occurs when <math|<frac|\<mathd\>\<Phi\>|\<mathd\>t>=0>,
giving
<\equation*>
0=Q<rsub|0> \<bbb-e\><rsup|-t<rsub|sat>/\<tau\><rsub|\<sigma\>>>+I<rsub|wind>*\<tau\><rsub|\<sigma\>>*<around*|(|1-\<bbb-e\><rsup|-t<rsub|sat>/\<tau\><rsub|\<sigma\>>>|)>
</equation*>
<\equation*>
t<rsub|sat>=\<tau\><rsub|\<sigma\>>*log<around*|(|1-<frac|Q<rsub|0>|I<rsub|wind>*\<tau\><rsub|\<sigma\>>>|)>
</equation*>
There is no saturation if
<\equation*>
<frac|Q<rsub|0>|I<rsub|wind>*\<tau\><rsub|\<sigma\>>>\<geqslant\>1
</equation*>
\;
Otherwise <math|\<tau\><rsub|\<sigma\>>> can be found by use of the product
log function (Wolfram Alpha),
<\equation*>
\<tau\><rsub|\<sigma\>>=<frac|1|<frac|Q<rsub|0>|I<rsub|wind>>-t<rsub|sat>*W<around*|(|<frac|I<rsub|wind
>t<rsub|sat>|Q<rsub|0>> exp<around*|(|<frac|I<rsub|wind
>t<rsub|sat>|Q<rsub|0>>|)>|)>>
</equation*>
\;
<paragraph|Oscillating field>
Assuming <math|\<mathd\>t\<gg\>\<tau\><rsub|c>>, the constant-wind charging
equation for a sphere is
<\equation*>
<wide|Q|\<dot\>>=-<frac|\<sigma\>|\<varepsilon\><rsub|0>>*Q+n*q
v<rsub|wind>*A<rsub|eff>
</equation*>
Replacing
<\equation*>
v<rsub|wind>\<rightarrow\>\<mu\>*E<rsub|ext>*cos<around*|(|\<omega\>*t|)>
</equation*>
gives
<\equation*>
<wide|Q|\<dot\>>=\<sigma\>*E<rsub|ext>*cos<around*|(|\<omega\>*t|)>*A<rsub|eff>-<frac|\<sigma\>|\<varepsilon\><rsub|0>>*Q
</equation*>
<\equation*>
<wide|Q|\<dot\>>=<frac|\<sigma\>|\<varepsilon\><rsub|0>><around*|(|\<varepsilon\><rsub|0>*E<rsub|ext>*cos<around*|(|\<omega\>*t|)>*A<rsub|eff>-Q|)>*
</equation*>
<\equation*>
<wide|Q|\<dot\>>=<frac|Q<rsub|ext>*cos<around*|(|\<omega\>*t|)>-Q|\<tau\><rsub|\<sigma\>>>
</equation*>
where
<\equation*>
Q<rsub|ext>\<equiv\>\<varepsilon\><rsub|0>*E<rsub|ext>*A<rsub|eff>
</equation*>
Assuming for now that <math|A<rsub|eff>> is independent of time (though we
know it is generally not, cf. uniform wind derivation), the solution is
[Wolfram Alpha]
<\equation*>
Q<around*|(|t|)>=<frac|Q<rsub|ext>*<around*|(|\<omega\>*\<tau\><rsub|\<sigma\>>*sin<around*|(|\<omega\>*t|)>+cos<around*|(|\<omega\>*t|)>|)>|1+\<omega\><rsup|2>*\<tau\><rsub|\<sigma\>><rsup|2>>+C*exp<around*|(|-<frac|t|\<tau\><rsub|\<sigma\>>>|)>
</equation*>
where <math|C> is an integration constant.
</body>
<\initial>
<\collection>
<associate|page-height|auto>
<associate|page-type|letter>
<associate|page-width|auto>
</collection>
</initial>
<\references>
<\collection>
<associate|auto-1|<tuple|1|?>>
<associate|auto-2|<tuple|2|?>>
<associate|auto-3|<tuple|3|?>>
<associate|auto-4|<tuple|4|?>>
<associate|auto-5|<tuple|5|?>>
<associate|auto-6|<tuple|6|?>>
<associate|auto-7|<tuple|7|?>>
<associate|auto-8|<tuple|8|?>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<with|par-left|<quote|4tab>|Isolated sphere
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.15fn>>
<with|par-left|<quote|4tab>|With constant windspeed
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Wake/collection-area correction
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Solving
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-4><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Regimes
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-5><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Conductivity from slope (no area
correction) <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Saturation potential
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7><vspace|0.15fn>>
</associate>
</collection>
</auxiliary>

164
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<TeXmacs|1.99.12>
<style|generic>
<\body>
<paragraph|Fundamental scales>
<\equation*>
R=object size
</equation*>
<with|color|pastel grey|<\equation*>
L=domain size
</equation*>>
<\equation*>
\<Phi\><rsub|0>=characteristic electric potential
</equation*>
<\equation*>
\<mu\>=mobility
</equation*>
<\equation*>
n=concentration
</equation*>
<\equation*>
\<mathd\>t=timestep
</equation*>
5 quantities, 3 units [m,s,C] <math|\<longrightarrow\>> 2 ratios
<paragraph|Dimensionless ratios>
<with|color|pastel brown|<\equation*>
\<frak-S\>=<frac|R|L>
</equation*>>
<\equation*>
\<frak-M\>\<equiv\><frac|v*\<mathd\>t|R>=<frac|\<mu\>*E*\<mathd\>t|R>=<frac|\<mu\>*\<Phi\><rsub|0>\<cdot\>\<mathd\>t|R\<cdot\>R>=<frac|\<mu\>*\<Phi\><rsub|0>*\<mathd\>t|R<rsup|2>>
</equation*>
<\equation*>
\<frak-N\>\<equiv\>n*R<rsup|3>
</equation*>
<paragraph|Conductivity>
Adding fundamental charge scale <math|e>,
<\equation*>
\<space\>\<sigma\>=<frac|j|E>=<frac|n*e*v|E>=<frac|n*e\<cdot\>\<mu\>*E|E>=n*e*\<mu\>
</equation*>
<\equation*>
\<frak-M\>=<frac|\<mu\>*\<Phi\><rsub|0>*\<mathd\>t|R<rsup|2>>=<frac|\<sigma\>*\<Phi\><rsub|0>*\<mathd\>t|n*e*R<rsup|2>>=<frac|\<sigma\>*\<Phi\><rsub|0>*\<mathd\>t|\<frak-N\>*e>*R
</equation*>
<\equation*>
\<frak-M\>*\<frak-N\>=<frac|\<sigma\>*\<Phi\><rsub|0>*\<mathd\>t|*e>*R=<frac|\<sigma\>*Q<rsub|0>*\<mathd\>t|*4*\<pi\>*\<varepsilon\><rsub|0>*R*e>*R\<propto\>*<frac|\<sigma\>*\<mathd\>t|\<varepsilon\><rsub|0>>*<frac|Q<rsub|0>|e>=<frac|\<mathd\>t|\<tau\>>*<frac|Q<rsub|0>|e>
</equation*>
where <math|\<tau\>=<frac|\<varepsilon\><rsub|0>|\<sigma\>>>. Slope of
relaxation probe trace is <math|slope=\<tau\><rsup|-1>=<frac|\<sigma\>|\<varepsilon\><rsub|0>>>.
<paragraph|Alternative (independent) ratios>
<\equation*>
\<cal-T\>\<equiv\><frac|\<mathd\>t|\<tau\>>
</equation*>
<\equation*>
\<cal-Q\>=<frac|Q<rsub|0>|e>
</equation*>
<\equation*>
\<cal-N\>\<equiv\>n*R<rsup|3>
</equation*>
<paragraph|Timestep constraint>
<\equation*>
v*\<mathd\>t\<less\>R
</equation*>
<\equation*>
\<mu\>*E*\<mathd\>t\<less\>R
</equation*>
<\equation*>
<frac|\<mu\>*\<Phi\><rsub|0>|R>*\<mathd\>t\<less\>R
</equation*>
<\equation*>
\<mathd\>t\<less\><frac|R<rsup|2>|\<mu\>*\<Phi\><rsub|0>>
</equation*>
<\equation*>
<frac|\<mathd\>t|\<tau\>>\<less\><frac|R<rsup|2>|\<tau\>*\<mu\>*\<Phi\><rsub|0>>
</equation*>
<\equation*>
<frac|\<mathd\>t|\<tau\>>\<less\><frac|\<sigma\>*R<rsup|2>|\<varepsilon\><rsub|0>*\<mu\>*\<Phi\><rsub|0>>
</equation*>
<\equation*>
<frac|\<mathd\>t|\<tau\>>\<less\><frac|n*e*\<mu\>*R<rsup|2>|\<varepsilon\><rsub|0>*\<mu\>*\<Phi\><rsub|0>>
</equation*>
<\equation*>
<frac|\<mathd\>t|\<tau\>>\<less\><frac|n*e*R<rsup|2>|\<varepsilon\><rsub|0>*\<Phi\><rsub|0>>
</equation*>
<\equation*>
\<Phi\>=\<Phi\><rsub|0>*exp<around*|(|<frac|t|\<tau\>>|)>=\<Phi\><rsub|0>*exp<around*|(|<frac|i*\<mathd\>t|\<tau\>>|)>=\<Phi\><rsub|0>*exp<around*|(|i*<frac|n*e*R<rsup|2>|\<varepsilon\><rsub|0>*\<Phi\><rsub|0>>|)>
</equation*>
\;
</body>
<\initial>
<\collection>
<associate|page-height|auto>
<associate|page-type|letter>
<associate|page-width|auto>
</collection>
</initial>
<\references>
<\collection>
<associate|auto-1|<tuple|1|?>>
<associate|auto-2|<tuple|2|?>>
<associate|auto-3|<tuple|3|?>>
<associate|auto-4|<tuple|4|?>>
<associate|auto-5|<tuple|5|?>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<with|par-left|<quote|4tab>|Fundamental scales
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Dimensionless ratios
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Conductivity
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Alternative (independent) ratios
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-4><vspace|0.15fn>>
<with|par-left|<quote|4tab>|Timestep constraint
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-5><vspace|0.15fn>>
</associate>
</collection>
</auxiliary>

82
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<TeXmacs|1.99.12>
<style|generic>
<\body>
<\equation*>
m*<frac|\<mathd\>v|\<mathd\>t>=q*E-m*\<nu\>*<around*|(|v-u|)>
</equation*>
<\equation*>
\<sigma\>=<frac|n*q<rsup|2>|m*\<nu\>>=n*q*\<mu\><application-space|1em>\<rightarrow\><application-space|1em>\<mu\>=<frac|q|m*\<nu\>><application-space|1em>\<rightarrow\><application-space|1em>m*\<nu\>=<frac|q|\<mu\>>
</equation*>
\;
<\equation*>
m*<frac|\<mathd\>v|\<mathd\>t>=q*<around*|(|E-<frac|v-u|\<mu\>>|)>
</equation*>
<\equation*>
<frac|\<mathd\>v|\<mathd\>t>=a*v+b
</equation*>
<\equation*>
a=-<frac|q|m*\<mu\>>
</equation*>
<\equation*>
b=<frac|q|m>*<around*|(|E+<frac|u|\<mu\>>|)>
</equation*>
The solution is
<\equation*>
v<around*|(|t|)>=C*exp<around*|(|a*t|)>-<frac|b|a>
</equation*>
<\equation*>
v<around*|(|t|)>=C*exp<around*|(|-<frac|q|m*\<mu\>>*t|)>-<frac|<frac|q|m>*<around*|(|E+<frac|u|\<mu\>>|)>|-<frac|q|m*\<mu\>>>=C*exp<around*|(|-<frac|t|\<tau\><rsub|c>>|)>+\<mu\>*E+u
</equation*>
where
<\equation*>
\<tau\><rsub|c>=<frac|m*\<mu\>|q>
</equation*>
The initial condition is
<\equation*>
v<around*|(|0|)>=C*+\<mu\>*E+u
</equation*>
<\equation*>
C=v<around*|(|0|)>-<around*|(|\<mu\>*E+u|)>
</equation*>
<\equation*>
v<around*|(|t|)>=v<around*|(|0|)>*exp<around*|(|-<frac|t|\<tau\><rsub|c>>|)>+<around*|(|\<mu\>*E+u|)><around*|(|1-exp<around*|(|-<frac|t|\<tau\><rsub|c>>|)>|)>
</equation*>
<\equation*>
v<around*|(|t+\<mathd\>t|)>=v<around*|(|t|)>*exp<around*|(|-<frac|\<mathd\>t|\<tau\><rsub|c>>|)>+<around*|(|\<mu\>*E+u|)><around*|(|1-exp<around*|(|-<frac|\<mathd\>t|\<tau\><rsub|c>>|)>|)>
</equation*>
The original timestep implemented by <with|font-shape|italic|Dragion> was
the <math|t\<rightarrow\>\<infty\>> limit,
<\equation*>
v<around*|(|t|)>=\<mu\>*E+u
</equation*>
\;
</body>
<\initial>
<\collection>
<associate|page-height|auto>
<associate|page-type|letter>
<associate|page-width|auto>
</collection>
</initial>

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Apache Commons Math includes the following code provided to the ASF under the
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has been imported from the Orekit space flight dynamics library.
===============================================================================
APACHE COMMONS MATH DERIVATIVE WORKS:
The Apache commons-math library includes a number of subcomponents
whose implementation is derived from original sources written
in C or Fortran. License terms of the original sources
are reproduced below.
===============================================================================
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the LevenbergMarquardtOptimizer class in package
org.apache.commons.math3.optimization.general
Original source copyright and license statement:
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EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
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===============================================================================
Copyright and license statement for the odex Fortran routine developed by
E. Hairer and G. Wanner and translated in GraggBulirschStoerIntegrator class
in package org.apache.commons.math3.ode.nonstiff:
Copyright (c) 2004, Ernst Hairer
Redistribution and use in source and binary forms, with or without
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===============================================================================
Copyright and license statement for the original Mersenne twister C
routines translated in MersenneTwister class in package
org.apache.commons.math3.random:
Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura,
All rights reserved.
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===============================================================================
The initial code for shuffling an array (originally in class
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inspired from the algorithm description provided in
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The textbook (containing a proof that the shuffle is uniformly random) is
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===============================================================================
License statement for the direction numbers in the resource files for Sobol sequences.
-----------------------------------------------------------------------------
Licence pertaining to sobol.cc and the accompanying sets of direction numbers
-----------------------------------------------------------------------------
Copyright (c) 2008, Frances Y. Kuo and Stephen Joe
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===============================================================================
The initial commit of the class "org.apache.commons.math3.special.BesselJ" is
an adapted version of code translated from the netlib Fortran program, rjbesl
http://www.netlib.org/specfun/rjbesl by R.J. Cody at Argonne National
Laboratory (USA). There is no license or copyright statement included with the
original Fortran sources.
===============================================================================
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and PowellOptimizer (package org.apache.commons.math3.optimization.general)
classes are based on the Python code in module "optimize.py" (version 0.5)
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(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
===============================================================================

9
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Apache Commons Math
Copyright 2001-2016 The Apache Software Foundation
This product includes software developed at
The Apache Software Foundation (http://www.apache.org/).
This product includes software developed for Orekit by
CS Systèmes d'Information (http://www.c-s.fr/)
Copyright 2010-2012 CS Systèmes d'Information

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Apache Commons Math 3.6.1 RELEASE NOTES
The Apache Commons Math team is pleased to announce the release of
commons-math3-3.6.1
The Apache Commons Math project is a library of lightweight, self-contained
mathematics and statistics components addressing the most common practical
problems not immediately available in the Java programming language or
commons-lang.
This is a minor bug fix release. It incorporates three bug fixes since version
3.6. The minimum version of the Java platform required to compile and use
Commons Math 3.6.1 is Java 5.
Changes in this version include:
Fixed Bugs:
o MATH-1316: Fix code example in User Guide > Statistics > Multiple linear regression.
o MATH-1342: Fix ODE integrators when multiple events are nearly simultaneous.
Changes:
o MATH-1317: Add uniformity unit tests to "RandomGeneratorAbstractTest".
For complete information on Apache Commons Math, including instructions on how
to submit bug reports, patches, or suggestions for improvement, see the Apache
Commons Math website:
http://commons.apache.org/proper/commons-math/

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function [phi,x,y,z] = diagread(filename)
fid = fopen(filename, 'rt');
dum = fscanf(fid, '%s', 1);
phi = str2num(fgetl(fid));
dum = fscanf(fid, '%s', 1);
x = str2num(fgetl(fid));
dum = fscanf(fid, '%s', 1);
y = str2num(fgetl(fid));
dum = fscanf(fid, '%s', 1);
z = str2num(fgetl(fid));
fclose(fid);

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package dragion.physics;
import java.util.Iterator;
import java.util.List;
import lestat.particles.Particle;
import mowgli.grid.CellScalar;
import mowgli.grid.Grid;
import mowgli.grid.InterpolationFactor;
import mowgli.shapes.Box;
public class CollisionalSinkUnipolar
{
final Grid grid;
final CellScalar sinkrate; // number / m^3 / s
final CellScalar density; // number / m^3
final CellScalar numberToLose; // number per cell; soy un perdedor so why don't you kill me...
public Grid getGrid()
{
return grid;
}
public CellScalar getSinkrate()
{
return sinkrate;
}
// public CellScalar getNumberToLose()
// {
// return numberToLose;
// }
public CellScalar getDensity()
{
return density;
}
public static final double alpha = 1e-12;
public CollisionalSinkUnipolar(Box bbox, int gridres)
{
this.grid = Grid.approximateSubdivide(bbox, gridres);
this.sinkrate = new CellScalar(grid, "sinkrate [#/m^3/s]");
this.density = new CellScalar(grid, "density [#/m^3]");
this.numberToLose = new CellScalar(grid, "numberToLose [#/timestep]");
}
public double checkPPC(double den, double np2c)
{
double cellvol = grid.getDx() * grid.getDy() * grid.getDz();
double ptclvol = np2c/den;
double ptclPerCell = cellvol/ptclvol;
return ptclPerCell;
}
public void collide(List<Particle> particles, double dt)
{
density.zero();
double[][][] density = this.density.getData();
Iterator<Particle> it = particles.iterator();
while (it.hasNext())
{
Particle p = it.next();
InterpolationFactor fac = grid.computeInterpolationFactor(p.getPos());
int i = fac.ilower;
int j = fac.jlower;
int k = fac.klower;
if (fac.isGood())
{
density[i][j][k] += p.getSpecies().getNp2c() / grid.getCellVolume(); // number density contribution per particle
}
else
{
throw new Error("Bad interpolation factor: position = " + p.getPos());
}
}
sinkrate.zero();
double[][][] sinkrate = this.sinkrate.getData();
//numberToLose.zero();
double[][][] numberToLose = this.numberToLose.getData();
for (int i = 0; i < grid.getNxc(); i++)
{
for (int j = 0; j < grid.getNyc(); j++)
{
for (int k = 0; k < grid.getNzc(); k++)
{
sinkrate[i][j][k] = density[i][j][k] * density[i][j][k] * alpha; // number / m^3 / s
numberToLose[i][j][k] += sinkrate[i][j][k] * dt * grid.getCellVolume();
}
}
}
it = particles.iterator();
while (it.hasNext())
{
Particle p = it.next();
InterpolationFactor fac = grid.computeInterpolationFactor(p.getPos());
int i = fac.ilower;
int j = fac.jlower;
int k = fac.klower;
if (fac.isGood())
{
if (numberToLose[i][j][k] > 0)
{
if (numberToLose[i][j][k] > p.getSpecies().getNp2c())
{
it.remove();
numberToLose[i][j][k] -= p.getSpecies().getNp2c();
}
else
{
if (Math.random() < numberToLose[i][j][k] / p.getSpecies().getNp2c())
{
it.remove();
numberToLose[i][j][k] -= p.getSpecies().getNp2c();
}
}
}
}
else
{
throw new Error("Bad interpolation factor: position = " + p.getPos());
}
}
}
}

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package dragion.physics;
import java.util.Collection;
import java.util.stream.Stream;
import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
import org.apache.commons.math3.special.Erf;
import lestat.particles.Advancer;
import lestat.particles.Particle;
import mowgli.Constants;
public class DriftDiffusionAdvancer implements Advancer
{
final double mobility;
final double temperature;
//final Vector3D windvel;
final boolean usetemp;
//final boolean usewind;
public DriftDiffusionAdvancer(double mobility, double temperature)//, Vector3D windvel)
{
this.mobility = mobility;
this.temperature = temperature;
// this.windvel = windvel;
this.usetemp = (mobility != 0) || (temperature != 0);
// this.usewind = !windvel.equals(Vector3D.ZERO);
}
@Override
public void advance(Collection<Particle> particles, double dt)
{
Stream<Particle> stream = particles.parallelStream();
stream.forEach((p) ->
{
//
// drift part
Vector3D vnew = p.getEfld().scalarMultiply(mobility);
//
// diffusion part... lots of Math.* method calls here so maybe some room for optimization
if (usetemp)
{
double D = mobility * Constants.boltzmann * temperature / Constants.qe; // diffusion coefficient
double lam = Math.sqrt(4 * D * dt); // characteristic diffusive jump length
double c = 2 * Math.random() - 1;
double dr = Math.pow(lam * Math.sqrt(Math.PI), 3) * Erf.erfInv(c);
double phi = 2 * Math.PI * Math.random();
double th = Math.acos(1 - 2 * Math.random());
double dxp = dr * Math.cos(phi) * Math.sin(th);
double dyp = dr * Math.sin(phi) * Math.sin(th);
double dzp = dr * Math.cos(th);
Vector3D jumplen = new Vector3D(dxp, dyp, dzp);
Vector3D vdiff = jumplen.scalarMultiply(1. / dt);
vnew = vnew.add(vdiff);
}
//
// wind advection part
// if (usewind)
// {
// vnew = vnew.add(windvel);
// }
//
p.setVel(vnew);
//
Vector3D xold = p.getPos();
p.setOldPos(xold);
//
Vector3D xnew = xold.add(vnew.scalarMultiply(dt));
p.setPos(xnew);
});
}
}

50
src/dragion/physics/FluidAdvancer.java Normal file
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package dragion.physics;
import java.util.Collection;
import java.util.stream.Stream;
import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
import lestat.particles.Advancer;
import lestat.particles.Particle;
public class FluidAdvancer implements Advancer
{
final Vector3D windvel;
final double mobility;
final double charge;
final double mass;
public FluidAdvancer(Vector3D windvel, double mobility, double charge, double mass)
{
this.windvel = windvel;
this.mobility = mobility;
this.charge = charge;
this.mass = mass;
}
@Override
public void advance(Collection<Particle> plist, double dt)
{
double tauc = mass * mobility / charge;
double expterm = Math.exp(-dt/tauc);
Stream<Particle> stream = plist.parallelStream();
stream.forEach((p) ->
{
Vector3D vold = p.getVel();
Vector3D euterm = p.getEfld().scalarMultiply(mobility).add(windvel);
Vector3D vnew = vold.scalarMultiply(expterm).add(euterm.scalarMultiply(1-expterm));
//
Vector3D xold = p.getPos();
Vector3D xnew = xold.add(vnew.scalarMultiply(dt));
//
p.setVel(vnew);
p.setOldPos(xold);
p.setPos(xnew);
});
}
}

106
src/dragion/physics/SideFluxInjector.java Normal file
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package dragion.physics;
import java.util.ArrayList;
import java.util.Collection;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.function.Function;
import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
import com.google.common.collect.ArrayListMultimap;
import com.google.common.collect.Multimap;
import lestat.particles.Particle;
import lestat.particles.Species;
import lestat.particles.emitcollect.Emitter;
import lestat.particles.emitcollect.RateDiscretizer;
import lestat.solvers.CompositeSolver;
import mowgli.samplers.QuadUniformSampler;
import mowgli.shapes.Box;
import mowgli.shapes.Box.Side;
import mowgli.shapes.Quad;
import mowgli.surfaces.EmbreeIntersector;
import mowgli.surfaces.Surface;
public class SideFluxInjector implements Emitter
{
final Box bbox;
final CompositeSolver solver;
final Species species;
final Function<Vector3D, Double> density;
final Function<Vector3D, Vector3D> windvel;
final double mobility;
Multimap<Side, Quad> allquads = ArrayListMultimap.create();
Map<Quad, QuadUniformSampler> samplers = new HashMap<>();
Map<Quad, RateDiscretizer> discretizers = new HashMap<>();
protected boolean checkInside(EmbreeIntersector intersector, Quad quad)
{
return (intersector.isInside(quad.getP1()) || intersector.isInside(quad.getP2()) || intersector.isInside(quad.getP3()) || intersector.isInside(quad.getP4()) || intersector.isInside(quad.getCenter()));
}
public SideFluxInjector(Box bbox, CompositeSolver solver, Species species, int subdiv, Function<Vector3D, Double> density, double mobility, EmbreeIntersector intersector, Function<Vector3D, Vector3D> windvel)
{
this.bbox = bbox;
this.solver = solver;
this.species = species;
this.density = density;
this.mobility = mobility;
this.windvel = windvel;
Map<Side, Quad> sides = Box.inwardPointingSides(bbox); // normals are inward pointing
for (Side side : sides.keySet())
{
Quad parent = sides.get(side);
// int subdiv = (int) Math.floor(Math.log(parent.getLength() / approxPanelSize) / Math.log(2)); // log2(parent length / panel size)
Quad[] children = Quad.subdivide(parent, subdiv);
for (Quad child : children)
{
if (!checkInside(intersector, child))
{
allquads.put(side, child);
samplers.put(child, new QuadUniformSampler(child));
discretizers.put(child, new RateDiscretizer());
}
}
}
}
@Override
public Collection<Particle> emit(double dt)
{
List<Particle> particles = new ArrayList<>();
for (Quad quad : allquads.values())
{
// estimate flux as (mobility * efld + windvel) * area at quad center
Vector3D ctr = quad.getCenter();
Vector3D ctrefld = solver.synthesizeElectricField(ctr);
double n0 = density.apply(ctr);
Vector3D ctrwindvel = windvel.apply(ctr);
Vector3D ctrvel = ctrefld.scalarMultiply(mobility).add(ctrwindvel);
double ctrvelprp = ctrvel.dotProduct(quad.getNormal());
if (ctrvelprp > 0) // normals are inward-pointing
{
RateDiscretizer discretizer = discretizers.get(quad);
discretizer.setRate(n0 * ctrvelprp * quad.getArea() / species.getNp2c());
int ntoemit = discretizer.update(dt);
//
for (int i = 0; i < ntoemit; i++)
{
double dtpart = dt * Math.random();
Vector3D oldpos = samplers.get(quad).get();
Vector3D newpos = oldpos.add(ctrvel.scalarMultiply(dtpart));
Particle p = new Particle(newpos, ctrvel, species);
p.setOldPos(oldpos);
particles.add(p);
}
}
}
return particles;
}
}

10
src/dragion/physics/SourceFunction.java Normal file
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package dragion.physics;
import java.util.function.BiFunction;
import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
public interface SourceFunction extends BiFunction<Vector3D, Double, Double>
{
}

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