diff --git a/.gitmodules b/.gitmodules
new file mode 100644
index 0000000000000000000000000000000000000000..81a1fbd21c98271e145cc16bc3cd41d3e9c93cbc
--- /dev/null
+++ b/.gitmodules
@@ -0,0 +1,3 @@
+[submodule "docs/hysop-doc"]
+ path = docs/hysop-doc
+ url = git@gricad-gitlab.univ-grenoble-alpes.fr:particle_methods/hysop-doc.git
diff --git a/docs/CMakeLists.txt b/docs/CMakeLists.txt
index 73790b9f77a200b2b4b2fffdbee56af77a00b1ff..658d7e19e70f6a93e3c105a50eb25171c49f4240 100644
--- a/docs/CMakeLists.txt
+++ b/docs/CMakeLists.txt
@@ -14,8 +14,6 @@ if(WITH_DOCUMENTATION)
endif()
execute_process(COMMAND head -n1 ${CMAKE_SOURCE_DIR}/NOTICE OUTPUT_VARIABLE HYSOP_COPYRIGHT)
- execute_process(COMMAND cat ${CMAKE_CURRENT_SOURCE_DIR}/sphinx/license-title.rst ${CMAKE_SOURCE_DIR}/LICENSE
- OUTPUT_FILE ${CMAKE_CURRENT_BINARY_DIR}/sphinx/license.rst)
# --------------------- Doxygen setup ---------------------
find_package(Doxygen)
@@ -58,22 +56,47 @@ if(WITH_DOCUMENTATION)
include(FindSphinxModule)
find_python_module(sphinx REQUIRED)
find_sphinx_module(sphinxcontrib bibtex REQUIRED)
-#import ISSUE: find_sphinx_module(sphinxcontrib doxylink REQUIRED)
+ #import ISSUE: find_sphinx_module(sphinxcontrib doxylink REQUIRED)
+
+ if(EXISTS ${CMAKE_CURRENT_SOURCE_DIR}/hysop-doc)
+ message("[Doc] Full documentation build (user + generated from source)")
+ set(HYSOP_DOC_SPHINX_SRC ${CMAKE_CURRENT_SOURCE_DIR}/hysop-doc)
+ set(BIBTEXFILE "hysop.bib")
+ configure_file (
+ "${HYSOP_DOC_SPHINX_SRC}/hysop.bib"
+ "${CMAKE_CURRENT_BINARY_DIR}/sphinx/hysop.bib" @ONLY)
+ # list of all dirs where rst or other sphinx required files are located
+ set(sphinx_doc_dirs
+ _static
+ _templates
+ devel
+ examples
+ install_guide
+ getting_started
+ reference
+ users_guide
+ figures
+ )
+ else()
+ message("[Doc] Minimal documentation generated from sources.")
+ message("[Doc] In order to build the full documentation consider runnig `git submodule init && git submodule update`")
+ set(HYSOP_DOC_SPHINX_SRC ${CMAKE_CURRENT_SOURCE_DIR}/sphinx_minimal)
+ set(BIBTEXFILE "")
+ # list of all dirs where rst or other sphinx required files are located
+ set(sphinx_doc_dirs _static reference )
+ endif()
+ message("[Doc] Sphinx source dir: ${HYSOP_DOC_SPHINX_SRC}")
# generate sphinx conf file
configure_file (
- "${CMAKE_CURRENT_SOURCE_DIR}/sphinx/conf.py.in"
+ "${HYSOP_DOC_SPHINX_SRC}/conf.py.in"
"${CMAKE_CURRENT_BINARY_DIR}/sphinx/conf.py" @ONLY)
# generate main sphinx index
configure_file (
- "${CMAKE_CURRENT_SOURCE_DIR}/sphinx/index.rst"
+ "${HYSOP_DOC_SPHINX_SRC}/index.rst"
"${CMAKE_CURRENT_BINARY_DIR}/sphinx/index.rst" @ONLY)
- configure_file (
- "${CMAKE_CURRENT_SOURCE_DIR}/sphinx/hysop.bib"
- "${CMAKE_CURRENT_BINARY_DIR}/sphinx/hysop.bib" @ONLY)
-
# --- Build sphinx targets ---
set(SPHINX_APIDOC sphinx-apidoc)
set(GENERATED_DOC ${CMAKE_CURRENT_BINARY_DIR}/sphinx/apidoc)
@@ -84,33 +107,24 @@ if(WITH_DOCUMENTATION)
${HYSOP_BUILD_PYTHONPATH}/hysop
)
- # list of all dirs where rst or other sphinx required files are located
- set(sphinx_doc_dirs
- _static
- _templates
- devel
- examples
- install_guide
- getting_started
- reference
- users_guide
- figures
- )
-
# symbolic links to directories of sphinx_doc_dirs, created when make apidoc is called.
foreach(dir ${sphinx_doc_dirs})
add_custom_command(
TARGET apidoc PRE_BUILD
- COMMAND ln -sf ${CMAKE_CURRENT_SOURCE_DIR}/sphinx/${dir}
+ COMMAND ln -sf ${HYSOP_DOC_SPHINX_SRC}/${dir}
${CMAKE_CURRENT_BINARY_DIR}/sphinx/)
endforeach()
- file(GLOB files_rst ${CMAKE_CURRENT_SOURCE_DIR}/sphinx/*.rst)
+ file(GLOB files_rst ${HYSOP_DOC_SPHINX_SRC}/*.rst)
foreach(filepath ${files_rst})
get_filename_component(filename ${filepath} NAME)
configure_file (
- "${CMAKE_CURRENT_SOURCE_DIR}/sphinx/${filename}"
+ "${HYSOP_DOC_SPHINX_SRC}/${filename}"
"${CMAKE_CURRENT_BINARY_DIR}/sphinx/${filename}" @ONLY)
endforeach()
+ ## Complete the license.rst with actual LICENCE file content
+ execute_process(COMMAND cat ${HYSOP_DOC_SPHINX_SRC}/license.rst ${CMAKE_SOURCE_DIR}/LICENSE
+ OUTPUT_FILE ${CMAKE_CURRENT_BINARY_DIR}/sphinx/license.rst)
+
# --> use make html or make latex ... to build documentation.
set(SPHINX_EXECUTABLE sphinx-build )
diff --git a/docs/config/hysop.doxyfile.in b/docs/config/hysop.doxyfile.in
index db08da3edde2751ff9b746d4988f570d7da9974f..61e83605ca5a478172e9cb765c2844576823d772 100644
--- a/docs/config/hysop.doxyfile.in
+++ b/docs/config/hysop.doxyfile.in
@@ -51,7 +51,7 @@ PROJECT_BRIEF = "Particle Methods simulation on hybrid architectures"
# pixels and the maximum width should not exceed 200 pixels. Doxygen will copy
# the logo to the output directory.
-PROJECT_LOGO = @CMAKE_CURRENT_SOURCE_DIR@/sphinx/figures/logo_hysop_nb.png
+PROJECT_LOGO = @CMAKE_CURRENT_SOURCE_DIR@/sphinx_minimal/_static/logo_hysop_couleur.png
# The OUTPUT_DIRECTORY tag is used to specify the (relative or absolute) path
# into which the generated documentation will be written. If a relative path is
diff --git a/docs/config/mainpage.doxygen b/docs/config/mainpage.doxygen
deleted file mode 100644
index 419d0ad3c934137e43d905b0024da12969164477..0000000000000000000000000000000000000000
--- a/docs/config/mainpage.doxygen
+++ /dev/null
@@ -1,175 +0,0 @@
-/** \mainpage HySoP
-\anchor doxyHySoP
-
-\section doxyHySoPIntro Introduction
-
-HySoP (Hybrid SimulatiOn with Particles) is a library dedicated to flow simulation based on particular methods, on hybrid (multi CPU-GPU) architecture.
-
-It is written in Python (high level functionnalities) and Fortran.
-
-The library is also supposed to provide a common framework to the following libraries :
-- the library of Adrien Magni (split3D and 2D for remeshing methods tests)
-- an interface to a spectral software from LEGI (Scales)
-- ...
-
-
-===================================================================================================================
-
-\section installhysop Install :
-
-\subsection reqInstall Requirements
-
-- a fortran compiler and a proper mpi implementation
-- fftw
-- cmake > 2.8
-- a python implementation including numpy and mpi4py.
-
-The install consists in 3 steps. First configuration of the package, makefile, setup.py and other files generation, then build of the underlying fortran libraries and of the python package and finally copy of the required files in the appropriate place.
-
-We denote :
-
-- SOURCEDIR as the directory which contains the sources (and this INSTALL file)
-- BUILDDIR the directory where the package will be configured and build
-- INSTALLDIR the directory where the package will be installed.
-Those 3 locations must be different.
-
-\subsection config Configuration :
-
-You need to generate some makefile and a setup.py that fits with your platform, your compilers,
-the libraries versions and so on. That will take place in BUILDDIR.
-
- -# get SOURCEDIR (i.e. download the package any way you want)
- -# create BUILDDIR. Any place, preferably local to your machine for efficiency reasons.
- -# enforce a fortran compiler using FC environment variable
- -# Change to BUILDDIR and run cmake
-
-\code
-mkdir $BUILDIR
-cd $BUILDDIR
-export FC=mpif90
-cmake $SOURCEDIR
-\endcode
-
-At the end of this step BUILDDIR contains all makefiles, setup.py and other required files for compilation.
-
-Some useful options for cmake :
-
-- -DFFTW_ROOT : where to find fftw if it's not in a "standard" place.
-- -DWITH_SCALES=ON/OFF : to compile an HySoP version including scales (default = on)
-- -DWITH_TESTS=ON/OFF: enable testing (i.e. prepare target "make test", default = off)
-
-example :
-\code
-mkdir /home/mylogin/buildHySoP
-cd /home/mylogin/buildHySoP
-export FC=mpif90
-module load cmake-2.8
-cmake -DFFTW_ROOT/softs/install/fftw3.1 ~/Softs/HySoP
-\endcode
-
-\subsection installDirConfig Install directory configuration :
-
-Installation scheme is following Python distutils one (<a href="http://docs.python.org/2/install/">http://docs.python.org/2/install/</a>).
-
-\subsubsection installDirConfigDefault Default configuration
-Default setup leads to a 'User scheme installation' : <a href="http://docs.python.org/2/install/#alternate-installation-the-user-scheme">http://docs.python.org/2/install/#alternate-installation-the-user-scheme</a>. Files are installed into subdirectories of Python 'site.USER_BASE' (USER_BASE/lib/pythonX.Y/site-packages for UNIX systems).
-
-After install completed, new files are located as follows :
- - python package in USER_BASE/lib/pythonX.Y/site-packages/hysop/
- - library in USER_BASE/lib/pythonX.Y/site-packages/hysop/lib
- - fortran modules in USER_BASE/lib/pythonX.Y/site-packages/hysop/include/Modules
- - CMake configuration files in USER_BASE/lib/pythonX.Y/site-packages/hysop/share/cmake
-
-\subsubsection installDirConfigCustom Custom configuration
-The other available setup is based on the Python 'Prefix scheme installation' : <a href="http://docs.python.org/2/install/#alternate-installation-unix-the-prefix-scheme">http://docs.python.org/2/install/#alternate-installation-unix-the-prefix-scheme</a>. You need to specify your prefix that replace the default USER_BASE. This prefix is passed to cmake through the variable PREFIX.
-
-Example :
-\code
-cmake -DPREFIX=/my/custom/prefix $SOURCEDIR
-\endcode
-After install completed, new files are located as follows :
- - python package in PREFIX/lib/pythonX.Y/site-packages/hysop/
- - library in PREFIX/lib/pythonX.Y/site-packages/hysop/lib
- - fortran modules in PREFIX/lib/pythonX.Y/site-packages/hysop/include/Modules
- - CMake configuration files in PREFIX/lib/pythonX.Y/site-packages/hysop/share/cmake
-
-
-\subsection buildhysop Build
-
-You need to build the underlying fortran libraries (mainly libhysop) and the python package.
-Just run "make" to do both.
-make python-build will run only python package building.
-
-\subsection installhysop Install
-
-Run :
-\code
-make python-install
-\endcode
-
-\subsection summaryInstall Summary
-
-Default process :
-\code
-mkdir $BUILDDIR
-cd $BUILDDIR
-export FC=mpif90
-cmake $SOURCEDIR
-make -jN
-make python-install
-\endcode
-
-Note : -jN for make command runs the build process on N processus. Choose a value of N that fits with your machine.
-
-Other useful commands :
-
-- make python-cleaninstall : to remove all installed files for the current build.
-- make clean : to clean the build (remove modules, .o ...)
-
-
-\subsection RemarkBrewedPython Note for OSX brewed python :
-If you are using a Python distribution from the <a href="http://brew.sh/">Homebrew</a> package manager for OSX, the --user install option is disbled. Default installation will fail "error: can't combine user with prefix, exec_prefix/home, or install_(plat)base". Installation require a prefix:
-\code
-cmake -DPREFIX=$(brew --prefix) $SOURCEDIR
-\endcode
-and it will install package to '$(brew --prefix)/lib/pythonX.Y/site-packages', for example '/usr/local/lib/python3.8/site-packages'
-
-A posible issue is that Python libraries and headers are not correctly discovered by CMake. We invite the user to carefully check that Python stuff in the cmake summary are correct and set the two variables : PYTHON_LIBRARY and PYTHON_INCLUDE_DIR.
-
-For example:
-\code
-cmake -DPYTHON_LIBRARY=$(brew --prefix)/Cellar/python/3.8.5/Frameworks/Python.framework/Versions/3.8/Python \
- -DPYTHON_INCLUDE_DIR=$(brew --prefix)/Cellar/python/3.8.5/Frameworks/Python.framework/Headers \
- $SOURCEDIR
-\endcode
-
-
-===================================================================================================================
-
-\section usehysop Use
-
-Use as a classical python package :
-
-\code
-import hysop as hy
-print hy.Box
-\endcode
-and so on ...
-
-Several examples are available in the directory "examples" of the tgz distribution.
-You can download them with "import examples.NAME"
-To get the list of available NAMES, try
-\code
-import examples
-help examples
-\endcode
-
-To run a parallel simulation try :
-\code
-mpirun -np 4 python ns3d.py
-\endcode
-
-
-
-
-*/
diff --git a/docs/hysop-doc b/docs/hysop-doc
new file mode 160000
index 0000000000000000000000000000000000000000..a09c2d1e571464d0e1368670b0a1a1cfe0736b2f
--- /dev/null
+++ b/docs/hysop-doc
@@ -0,0 +1 @@
+Subproject commit a09c2d1e571464d0e1368670b0a1a1cfe0736b2f
diff --git a/docs/sphinx/_static/layout.css b/docs/sphinx/_static/layout.css
deleted file mode 100644
index edaac746775b919935695120396b89c31cc17921..0000000000000000000000000000000000000000
--- a/docs/sphinx/_static/layout.css
+++ /dev/null
@@ -1,37 +0,0 @@
-@media (max-width:1000px) {
- #sidebar { display: none; }
-}
-
-/* Background color for sidebar and navbar */
-:root {
- --menu-bg-color: #f1f6d0;
-}
-
-
-#sidebar {
- padding-left: 10px;
- background-color: var(--menu-bg-color);
-}
-
-/* body { */
-/* padding-top: 120px */
-/* } */
-
-.navbar-default {
- background-color: var(--menu-bg-color);
- border-color: var(--menu-bg-color);
-}
-
-/* To remove project name displayed under logo*/
-a.navbar-brand{
- color: transparent!important;
-}
-
-/* index gallery scrolled or not ...*/
-.index-gallery {
- overflow-x: scroll;
- overflow-y: hidden;
- white-space: nowrap;
- width: 100%;
- margin-bottom: 1cm;
-}
diff --git a/docs/sphinx/_static/logo_ljk_blanc2.png b/docs/sphinx/_static/logo_ljk_blanc2.png
deleted file mode 100644
index 944016af8a56acfaa537e86424fc6a55b3bc2b66..0000000000000000000000000000000000000000
Binary files a/docs/sphinx/_static/logo_ljk_blanc2.png and /dev/null differ
diff --git a/docs/sphinx/_templates/layout.html b/docs/sphinx/_templates/layout.html
deleted file mode 100644
index 0ae7c77449d8ed02f432c764d3a985825e2bd85f..0000000000000000000000000000000000000000
--- a/docs/sphinx/_templates/layout.html
+++ /dev/null
@@ -1,5 +0,0 @@
-{# Import the theme's layout. #}
-{% extends "!layout.html" %}
-
-{# Custom CSS overrides #}
-{% set bootswatch_css_custom = ['_static/layout.css'] %}
diff --git a/docs/sphinx/_templates/menu.html b/docs/sphinx/_templates/menu.html
deleted file mode 100644
index aedcfc490910cfb8c2fdbee7404330e92fa3cea5..0000000000000000000000000000000000000000
--- a/docs/sphinx/_templates/menu.html
+++ /dev/null
@@ -1,8 +0,0 @@
-<h3><a href="{{ pathto(master_doc) }}">Table Of Contents</a></h3>
-{{ toctree(includehidden=True) }}
-
-<h3>Links</h3>
-<p><a class="reference external" href="https://gitlab.com/LJK/hysop" target="_top">HySoP on GitLab</a></p>
-<p><a class="reference external" href="http://my.cdash.org/index.php?project=HySoP" target="_top">HySoP dashboard</a></p>
-
-<p><a href="http://ljk.imag.fr/"><img src="{{ pathto('_static/logo_ljk_blanc2.png', 2) }}" " /></a></p>
diff --git a/docs/sphinx/devel/about_codegen.rst b/docs/sphinx/devel/about_codegen.rst
deleted file mode 100644
index 9826a6d38bde2776222ec885b738b5e92b34713c..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/about_codegen.rst
+++ /dev/null
@@ -1,12 +0,0 @@
-.. _about_codegen:
-
-Code generation and autotuning
-==============================
-
-
-
-To Be done
-
-see
-:class:`~hysop.backend.device.autotunable_kernel.AutotunableKernel`
- and hysop.symbolic
diff --git a/docs/sphinx/devel/about_doc.rst b/docs/sphinx/devel/about_doc.rst
deleted file mode 100644
index 94e7b030389d6e243056dd0bcdf55bd2113eb9a6..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/about_doc.rst
+++ /dev/null
@@ -1,16 +0,0 @@
-.. _about_doc:
-
-How to document classes in HySoP
-=================================
-
-We try to fit as much as possible with `numpydoc guide <https://numpydoc.readthedocs.io/en/latest/>`_
-guide standards.
-
-See the files fields/continuous.py and operator/drag_and_lift.py which
-are used as reference files for documentation.
-
-Mind the need of `raw strings litterals <https://docs.python.org/3/reference/lexical_analysis.html#string-and-bytes-literals>`_ in python docstrings for keeping the `\stuff` from math
-environments `".. :math:"`.
-
-Mind the 'space' character before colon character `":"` in parameters or
-attributes definitions.
diff --git a/docs/sphinx/devel/about_operators_graph.rst b/docs/sphinx/devel/about_operators_graph.rst
deleted file mode 100644
index 2ba3dfd61b7a4f6f31c06ef99f8f2ac5c2bb0600..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/about_operators_graph.rst
+++ /dev/null
@@ -1,45 +0,0 @@
-.. _about_operators_graph:
-
-How HySoP deals with computational graphs of operators
-======================================================
-
-.. inheritance-diagram::
- hysop.core.graph.computational_node.ComputationalGraphNode
- hysop.core.graph.computational_graph.ComputationalGraph
- hysop.core.graph.node_generator.ComputationalGraphNodeGenerator
- hysop.core.graph.computational_operator.ComputationalGraphOperator
- hysop.core.graph.computational_node_frontend.ComputationalGraphNodeFrontend
- hysop.operator.directional.directional.DirectionalOperatorBase
- hysop.operator.directional.directional.DirectionalOperatorGenerator
- hysop.operator.directional.directional.DirectionalOperatorFrontend
- hysop.numerics.splitting.directional_splitting.DirectionalSplitting
- hysop.problem.Problem
- :parts: 1
- :caption: Inheritance diagram for HySoP computational graphs.
-
-
-HySoP library is designed for the user to manipulate only operators
-and problems. It internally build a directed graph of operators (low level
-computations or data operations) for running a single a solver
-iteration. Therefore, solving a Problem means running iteratively the
-graph traversal. Order of inserting elements in the Problem using
-:meth:`~hysop.problem.Problem.insert` (or graph using :meth:`~hysop.core.graph.computational_graph.ComputationalGraph.push_nodes`)
-is important.
-
-Two kind of object can be inserted into a graph:
-
-- a :class:`~hysop.core.graph.computational_node.ComputationalGraphNode` by means of:
-
- * an operator (:class:`~hysop.core.graph.computational_operator.ComputationalGraphOperator`)
- * a sub-graph (:class:`~hysop.core.graph.computational_graph.ComputationalGraph`) when using a operator of operators
-
-- an operator generator (:class:`~hysop.core.graph.node_generator.ComputationalGraphNodeGenerator` or
- :class:`~hysop.core.graph.computational_node_frontend.ComputationalGraphNodeFrontend`). In that case, only generated
- operators are inserted into graph, the generator itself is not inserted.
-
-We distinguish directional operators from operators. Directional
-operators must split into several operators (at least one per spatial
-directions) using a :class:`~hysop.numerics.splitting.directional_splitting.DirectionalSplitting`. The nested
-operators are sharing a common basis:
-:class:`~hysop.operator.directional.directional.DirectionalOperatorBase` that are generated by :class:`~hysop.operator.directional.directional.DirectionalOperatorGenerator`. Directional operators generators will automatically insert also
-transposition operators (see :ref:`about_work_arrays`).
diff --git a/docs/sphinx/devel/about_work_arrays.rst b/docs/sphinx/devel/about_work_arrays.rst
deleted file mode 100644
index 64de4b2e00c8fa649d7eac2974e657558363f629..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/about_work_arrays.rst
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _about_work_arrays:
-
-Temporary work arays in HySoP
-=============================
-
-
-TO BE DONE
-
-See :class:`~hysop.core.memory.memory_request.MemoryRequest`
diff --git a/docs/sphinx/devel/coding_guidelines.rst b/docs/sphinx/devel/coding_guidelines.rst
deleted file mode 100644
index 2ae3068d6772dee24c68e59c43250901a3965ae2..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/coding_guidelines.rst
+++ /dev/null
@@ -1,11 +0,0 @@
-.. _coding_guidelines:
-
-How to write proper code in HySoP
-=================================
-
-Developper should follow python recommandations for code style.
-
-Lower-level functions should be tested in the appropriate unitary
-tests folders.
-
-Higher-level functionalities should be exposed in an example file.
diff --git a/docs/sphinx/devel/fortran2python.rst b/docs/sphinx/devel/fortran2python.rst
deleted file mode 100644
index 13141dcd676fcce1048b8624adc8ed5653a558ea..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/fortran2python.rst
+++ /dev/null
@@ -1,32 +0,0 @@
-.. _fortran2python:
-
-Fortran to python interface
-===========================
-
-This page propose a tutorial to call some fortran subroutines in python within hysop.
-
-
-Sources:
-
- * some fortran files which contain some subroutines inside some modules
- * a pyf file
-
-
-
- TO BE DONE !!!
-
-
-
-
-
-Kind values
------------
-
-According to f2py doc: "Currently, F2PY can handle only <type spec>(kind=<kindselector>) declarations where <kindselector> is a numeric integer (e.g. 1, 2, 4,...) but not a function call KIND(..) or any other expression. F2PY needs to know what would be the corresponding C type and a general solution for that would be too complicated to implement."
-
-To overcome this, a .f2py_f2cmap is generated during the build, from hysop/f2py_f2cmap.in.
-It contains all the required mappings defined as a dictionnary, like::
-
- {'integer':{'c_int':'int'}, 'real':{'real64':'double', 'wp':'double'}}
-
-In that example, a fortran :code:'real(kind=wp)' will be mapped into a double-precision real in python.
diff --git a/docs/sphinx/devel/hysop_validation.rst b/docs/sphinx/devel/hysop_validation.rst
deleted file mode 100644
index 118e89aa24892cacffadedeae7a4e1e208f22797..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/hysop_validation.rst
+++ /dev/null
@@ -1,7 +0,0 @@
-.. _hysop_validation:
-
-
-HySoP validation
-================
-
-To be done
diff --git a/docs/sphinx/devel/index.rst b/docs/sphinx/devel/index.rst
deleted file mode 100644
index a867ba0ea7c3c9f198ac465200106782157617db..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/index.rst
+++ /dev/null
@@ -1,20 +0,0 @@
-.. _devel:
-
-
-############################
-HySoP Developers Guide
-############################
-
-This part is for developers of Hysop and provides guidelines on the "proper" ways to write python code, documentation, examples, tests ... in HySoP.
-
-.. toctree::
- :maxdepth: 2
-
- about_doc
- coding_guidelines
- fortran2python
- about_operators_graph
- about_work_arrays
- about_codegen
- memo_sphinx
- hysop_validation
diff --git a/docs/sphinx/devel/memo_sphinx.rst b/docs/sphinx/devel/memo_sphinx.rst
deleted file mode 100644
index eaf7387b6a1f3bd815ba5eac74e992ec97f23c19..0000000000000000000000000000000000000000
--- a/docs/sphinx/devel/memo_sphinx.rst
+++ /dev/null
@@ -1,89 +0,0 @@
-.. _memo_sphinx:
-
-Reminder for sphinx utilities
-=============================
-
-Source code for this page::
-
- To print source Code::
-
- for i in range(3):
- do un_truc()
-
-
- Hyperlinks `HySoP Web`_.
-
-
- .. math::
-
- \begin{eqnarray}
- a &=& \frac{1}{2}\\
- \alpha &=& 3 \times \nabla \omega
- \end{eqnarray}
-
- To print a link to some HySoP object:
-
- .. currentmodule:: hysop.operator.discrete
-
- An example of a discrete operator, :class:`stretching.Conservative`, used to represent the
- stretching part of NS.
-
- Reference to :ref:`forces`.
-
-
- Links
- -----
-
- * rst :
- http://sphinx-doc.org/rest.html
-
- * math support
- http://sphinx-doc.org/ext/math.html
-
- * numpydoc :
- https://github.com/numpy/numpy/blob/master/doc/HOWTO_DOCUMENT.rst.txt
-
- .. _Hysop Web: https://forge.imag.fr
-
-
-Which results in:
-
-To print source Code::
-
- for i in range(3):
- do un_truc()
-
-
-Hyperlinks `HySoP Web`_.
-
-
-.. math::
- \begin{eqnarray}
- a &=& \frac{1}{2}\\
- \alpha &=& 3 \times \nabla \omega
- \end{eqnarray}
-
-To print a link to some HySoP object:
-
-.. currentmodule:: hysop.operator.discrete
-
-An example of a discrete operator, :class:`stretching.Conservative`, used to represent the
-stretching part of NS.
-
-Reference to :ref:`forces`.
-
-
-Links
------
-
- * rst :
- http://sphinx-doc.org/rest.html
-
- * math support
- http://sphinx-doc.org/ext/math.html
-
- * numpydoc :
- https://github.com/numpy/numpy/blob/master/doc/HOWTO_DOCUMENT.rst.txt
-
-
-.. _Hysop Web: https://forge.imag.fr
diff --git a/docs/sphinx/examples/index.rst b/docs/sphinx/examples/index.rst
deleted file mode 100644
index a712ddedb667b2255e8f28c4dc1599df803c45ba..0000000000000000000000000000000000000000
--- a/docs/sphinx/examples/index.rst
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _examples:
-
-############################
-HySoP Examples
-############################
-
-Some tutorials and examples:
-- on how to use HySoP
-- demos about HySoP perf and capabilities
diff --git a/docs/sphinx/figures/PolesExample.png b/docs/sphinx/figures/PolesExample.png
deleted file mode 100644
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diff --git a/docs/sphinx/figures/PorousHemiSphereExample.png b/docs/sphinx/figures/PorousHemiSphereExample.png
deleted file mode 100644
index 5c713bfe88cf8762ba2ae9528a6204911bf0dff3..0000000000000000000000000000000000000000
Binary files a/docs/sphinx/figures/PorousHemiSphereExample.png and /dev/null differ
diff --git a/docs/sphinx/figures/QuadriPoleExample.png b/docs/sphinx/figures/QuadriPoleExample.png
deleted file mode 100644
index 53fe705f4a059abd694d02a19d6e475d28fa6bf1..0000000000000000000000000000000000000000
Binary files a/docs/sphinx/figures/QuadriPoleExample.png and /dev/null differ
diff --git a/docs/sphinx/figures/cluster_general.jpg b/docs/sphinx/figures/cluster_general.jpg
deleted file mode 100644
index 82d1004bac7e2226e8610b9be31009d6a84c0e64..0000000000000000000000000000000000000000
Binary files a/docs/sphinx/figures/cluster_general.jpg and /dev/null differ
diff --git a/docs/sphinx/figures/cluster_general.pdf b/docs/sphinx/figures/cluster_general.pdf
deleted file mode 100644
index 072bca5845185ff382a10d1fa5d4b8a72c467ba8..0000000000000000000000000000000000000000
Binary files a/docs/sphinx/figures/cluster_general.pdf and /dev/null differ
diff --git a/docs/sphinx/figures/comm_split.graffle b/docs/sphinx/figures/comm_split.graffle
deleted file mode 100644
index 698d9951a9efa74cd04f6e9fecbce12629b68eff..0000000000000000000000000000000000000000
--- a/docs/sphinx/figures/comm_split.graffle
+++ /dev/null
@@ -1,851 +0,0 @@
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- <key>a</key>
- <string>0.4</string>
- <key>b</key>
- <string>0</string>
- <key>g</key>
- <string>0</string>
- <key>r</key>
- <string>0</string>
- </dict>
- <key>Fuzziness</key>
- <real>0.0</real>
- <key>ShadowVector</key>
- <string>{0, 2}</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>a</key>
- <string>0.75</string>
- <key>b</key>
- <string>0</string>
- <key>g</key>
- <string>0</string>
- <key>r</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <key>TextRelativeArea</key>
- <string>{{0.125, 0.25}, {0.75, 0.5}}</string>
- <key>isConnectedShape</key>
- <true/>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{253.62018038332462, 118.40337514877319}, {107, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>252</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- </dict>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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-
-\f0\fs24 \cf0 MPI distribution}</string>
- </dict>
- </dict>
- </array>
- <key>ID</key>
- <integer>248</integer>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>228</integer>
- <key>Points</key>
- <array>
- <string>{18.625, 140}</string>
- <string>{504, 141}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>0</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>9</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>230</integer>
- <key>Points</key>
- <array>
- <string>{161, 94.75}</string>
- <string>{162, 354.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>0</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>232</integer>
- <key>Points</key>
- <array>
- <string>{341.93817138671875, 99.5}</string>
- <string>{374.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>233</integer>
- <key>Points</key>
- <array>
- <string>{315.93817138671875, 99.5}</string>
- <string>{348.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>234</integer>
- <key>Points</key>
- <array>
- <string>{289.93817138671875, 99.5}</string>
- <string>{322.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>235</integer>
- <key>Points</key>
- <array>
- <string>{263.93817138671875, 99.5}</string>
- <string>{296.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>236</integer>
- <key>Points</key>
- <array>
- <string>{159.93817138671875, 99.5}</string>
- <string>{192.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>237</integer>
- <key>Points</key>
- <array>
- <string>{237.93817138671875, 99.5}</string>
- <string>{270.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>238</integer>
- <key>Points</key>
- <array>
- <string>{211.93817138671875, 99.5}</string>
- <string>{244.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>239</integer>
- <key>Points</key>
- <array>
- <string>{185.93817138671875, 99.5}</string>
- <string>{218.93817138671875, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{151, 97}, {303, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>240</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- </dict>
- <key>Text</key>
- <dict>
- <key>Align</key>
- <integer>0</integer>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{394.625, 64.5}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>241</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- </dict>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{138.93817138671875, 64.5}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>242</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- </dict>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- <dict>
- <key>Bounds</key>
- <string>{{471.93817138671875, 75.000000000000227}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>243</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
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- </dict>
- <key>Text</key>
- <dict>
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- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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-\f0\fs24 \cf0 x}</string>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>244</integer>
- <key>Points</key>
- <array>
- <string>{69, 100}</string>
- <string>{512, 99.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>0</string>
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- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>245</integer>
- <key>Points</key>
- <array>
- <string>{425.93817138671875, 94.75}</string>
- <string>{426.93817138671875, 354.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>0</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
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- <dict>
- <key>Bounds</key>
- <string>{{259, 350}, {231, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>201</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
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- </dict>
- <key>Text</key>
- <dict>
- <key>Align</key>
- <integer>0</integer>
- <key>Text</key>
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- <dict>
- <key>Bounds</key>
- <string>{{177.625, 274.75}, {179, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>200</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
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- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
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- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
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- <key>Text</key>
- <dict>
- <key>Align</key>
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- <dict>
- <key>Bounds</key>
- <string>{{193.5, 319.5}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>199</integer>
- <key>Shape</key>
- <string>Circle</string>
- <key>Style</key>
- <dict/>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- <key>VerticalPad</key>
- <integer>0</integer>
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- <dict>
- <key>Bounds</key>
- <string>{{116, 244.25}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>198</integer>
- <key>Shape</key>
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- <dict>
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- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- <key>VerticalPad</key>
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- <dict>
- <key>Bounds</key>
- <string>{{31, 186}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>28</integer>
- <key>Shape</key>
- <string>Circle</string>
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- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
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- <key>VerticalPad</key>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <integer>178</integer>
- <key>Points</key>
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- <string>{368.625, 337}</string>
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- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
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- <true/>
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- <dict>
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- <string>LineGraphic</string>
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- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
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- <key>r</key>
- <string>1</string>
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- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <integer>180</integer>
- <key>Points</key>
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- <string>{342.625, 337}</string>
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- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
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- <dict>
- <key>Class</key>
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- <dict>
- <key>stroke</key>
- <dict>
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- <string>FilledBall</string>
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- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <string>{689.625, 207.875}</string>
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- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
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- <true/>
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- <dict>
- <key>stroke</key>
- <dict>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <key>Points</key>
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- <dict>
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- <dict>
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- <dict>
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- <string>LineGraphic</string>
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- <dict>
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- <dict>
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- <true/>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <key>Points</key>
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- <dict>
- <key>stroke</key>
- <dict>
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- <string>FilledBall</string>
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- <true/>
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- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
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- <dict>
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- <dict>
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- <dict>
- <key>Class</key>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <dict>
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- <string>LineGraphic</string>
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- <key>Points</key>
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- <dict>
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- <dict>
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- <dict>
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- <string>LineGraphic</string>
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- <dict>
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- <dict>
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- <dict>
- <key>fill</key>
- <dict>
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- <dict>
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- <key>stroke</key>
- <dict>
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- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>Group</string>
- <key>Graphics</key>
- <array>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>271</integer>
- <key>Points</key>
- <array>
- <string>{687.625, 143.875}</string>
- <string>{687.625, 207.875}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.226548</string>
- <key>g</key>
- <string>0.727356</string>
- <key>r</key>
- <string>1</string>
- </dict>
- <key>HeadArrow</key>
- <string>0</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>FilledArrow</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>272</integer>
- <key>Points</key>
- <array>
- <string>{658.8125, 143.875}</string>
- <string>{658.8125, 207.875}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.226548</string>
- <key>g</key>
- <string>0.727356</string>
- <key>r</key>
- <string>1</string>
- </dict>
- <key>HeadArrow</key>
- <string>0</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>FilledArrow</string>
- </dict>
- </dict>
- </dict>
- </array>
- <key>ID</key>
- <integer>270</integer>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>273</integer>
- <key>Points</key>
- <array>
- <string>{635.4375, 143.875}</string>
- <string>{635.4375, 207.875}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.226548</string>
- <key>g</key>
- <string>0.727356</string>
- <key>r</key>
- <string>1</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>274</integer>
- <key>Points</key>
- <array>
- <string>{606.625, 143.875}</string>
- <string>{606.625, 207.875}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.226548</string>
- <key>g</key>
- <string>0.727356</string>
- <key>r</key>
- <string>1</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>Pattern</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>Head</key>
- <dict>
- <key>ID</key>
- <integer>276</integer>
- <key>Info</key>
- <integer>2</integer>
- </dict>
- <key>ID</key>
- <integer>275</integer>
- <key>Points</key>
- <array>
- <string>{682.21875, 130}</string>
- <string>{655.625, 112}</string>
- <string>{635.4375, 129.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.348234</string>
- <key>g</key>
- <string>1</string>
- <key>r</key>
- <string>0.39397</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>LineType</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>276</integer>
- <key>Points</key>
- <array>
- <string>{655.625, 130}</string>
- <string>{649.8125, 117.5}</string>
- <string>{635.4375, 129.5}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.348234</string>
- <key>g</key>
- <string>1</string>
- <key>r</key>
- <string>0.39397</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>LineType</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>Head</key>
- <dict>
- <key>ID</key>
- <integer>280</integer>
- </dict>
- <key>ID</key>
- <integer>277</integer>
- <key>Points</key>
- <array>
- <string>{584.625, 132}</string>
- <string>{610.625, 111}</string>
- <string>{626.672567876833, 129.86911818006573}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.348234</string>
- <key>g</key>
- <string>1</string>
- <key>r</key>
- <string>0.39397</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>LineType</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>278</integer>
- <key>Points</key>
- <array>
- <string>{609.625, 132}</string>
- <string>{616.9375, 119.25}</string>
- <string>{628.5625, 130.25}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.348234</string>
- <key>g</key>
- <string>1</string>
- <key>r</key>
- <string>0.39397</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledArrow</string>
- <key>Legacy</key>
- <true/>
- <key>LineType</key>
- <integer>1</integer>
- <key>TailArrow</key>
- <string>0</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>279</integer>
- <key>Points</key>
- <array>
- <string>{603.625, 137.75}</string>
- <string>{636.625, 137.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{625.12500011175871, 130.25}, {16.5, 15}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>280</integer>
- <key>Line</key>
- <dict>
- <key>ID</key>
- <integer>269</integer>
- <key>Position</key>
- <real>0.11363636702299118</real>
- <key>RotationType</key>
- <integer>0</integer>
- </dict>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.398614</string>
- <key>g</key>
- <string>1</string>
- <key>r</key>
- <string>0.412761</string>
- </dict>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{517.625, 150.75}, {179, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>282</integer>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>shadow</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- <key>stroke</key>
- <dict>
- <key>Draws</key>
- <string>NO</string>
- </dict>
- </dict>
- <key>Text</key>
- <dict>
- <key>Align</key>
- <integer>0</integer>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
-\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
-{\colortbl;\red255\green255\blue255;\red55\green180\blue29;}
-\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural
-
-\f0\fs24 \cf2 0 1 2 3 4 5 6 \
-\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural
-\cf0 1 2 3 4 5 6 7}</string>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{533.5, 195.5}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>283</integer>
- <key>Shape</key>
- <string>Circle</string>
- <key>Style</key>
- <dict/>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
-\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
-{\colortbl;\red255\green255\blue255;}
-\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
-
-\f0\fs24 \cf0 P2}</string>
- <key>VerticalPad</key>
- <integer>0</integer>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{456, 120.25}, {54, 36}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>284</integer>
- <key>Shape</key>
- <string>Circle</string>
- <key>Style</key>
- <dict/>
- <key>Text</key>
- <dict>
- <key>Text</key>
- <string>{\rtf1\ansi\ansicpg1252\cocoartf1348\cocoasubrtf170
-\cocoascreenfonts1{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
-{\colortbl;\red255\green255\blue255;}
-\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\pardirnatural\qc
-
-\f0\fs24 \cf0 P1}</string>
- <key>VerticalPad</key>
- <integer>0</integer>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>291</integer>
- <key>Points</key>
- <array>
- <string>{525.625, 137.75}</string>
- <string>{558.625, 137.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0</string>
- <key>g</key>
- <string>0.00927087</string>
- <key>r</key>
- <string>1</string>
- </dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>292</integer>
- <key>Points</key>
- <array>
- <string>{577.625, 137.75}</string>
- <string>{610.625, 137.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Class</key>
- <string>LineGraphic</string>
- <key>ID</key>
- <integer>293</integer>
- <key>Points</key>
- <array>
- <string>{551.625, 137.75}</string>
- <string>{584.625, 137.75}</string>
- </array>
- <key>Style</key>
- <dict>
- <key>stroke</key>
- <dict>
- <key>HeadArrow</key>
- <string>FilledBall</string>
- <key>Legacy</key>
- <true/>
- <key>TailArrow</key>
- <string>FilledBall</string>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{572.99999998509884, 130.25}, {16.5, 15}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>294</integer>
- <key>Line</key>
- <dict>
- <key>ID</key>
- <integer>292</integer>
- <key>Position</key>
- <real>0.10984848439693451</real>
- <key>RotationType</key>
- <integer>0</integer>
- </dict>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.357603</string>
- <key>g</key>
- <string>0.8226</string>
- <key>r</key>
- <string>1</string>
- </dict>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{598.99999998509884, 130.25}, {16.5, 15}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>295</integer>
- <key>Line</key>
- <dict>
- <key>ID</key>
- <integer>279</integer>
- <key>Position</key>
- <real>0.10984848439693451</real>
- <key>RotationType</key>
- <integer>0</integer>
- </dict>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.357603</string>
- <key>g</key>
- <string>0.8226</string>
- <key>r</key>
- <string>1</string>
- </dict>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{650.99999928474426, 130.25}, {16.5, 15}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>296</integer>
- <key>Line</key>
- <dict>
- <key>ID</key>
- <integer>269</integer>
- <key>Position</key>
- <real>0.89772725105285645</real>
- <key>RotationType</key>
- <integer>0</integer>
- </dict>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.357603</string>
- <key>g</key>
- <string>0.8226</string>
- <key>r</key>
- <string>1</string>
- </dict>
- </dict>
- </dict>
- </dict>
- <dict>
- <key>Bounds</key>
- <string>{{676.99999928474426, 130.25}, {16.5, 15}}</string>
- <key>Class</key>
- <string>ShapedGraphic</string>
- <key>ID</key>
- <integer>297</integer>
- <key>Line</key>
- <dict>
- <key>ID</key>
- <integer>268</integer>
- <key>Position</key>
- <real>0.89772725105285645</real>
- <key>RotationType</key>
- <integer>0</integer>
- </dict>
- <key>Shape</key>
- <string>Rectangle</string>
- <key>Style</key>
- <dict>
- <key>fill</key>
- <dict>
- <key>Color</key>
- <dict>
- <key>b</key>
- <string>0.357603</string>
- <key>g</key>
- <string>0.8226</string>
- <key>r</key>
- <string>1</string>
- </dict>
- </dict>
- </dict>
- </dict>
- </array>
- <key>GridInfo</key>
- <dict/>
- <key>GuidesLocked</key>
- <string>NO</string>
- <key>GuidesVisible</key>
- <string>YES</string>
- <key>HPages</key>
- <integer>2</integer>
- <key>ImageCounter</key>
- <integer>1</integer>
- <key>KeepToScale</key>
- <false/>
- <key>Layers</key>
- <array>
- <dict>
- <key>Lock</key>
- <string>NO</string>
- <key>Name</key>
- <string>Calque 1</string>
- <key>Print</key>
- <string>YES</string>
- <key>View</key>
- <string>YES</string>
- </dict>
- </array>
- <key>LayoutInfo</key>
- <dict>
- <key>Animate</key>
- <string>NO</string>
- <key>circoMinDist</key>
- <real>18</real>
- <key>circoSeparation</key>
- <real>0.0</real>
- <key>layoutEngine</key>
- <string>dot</string>
- <key>neatoSeparation</key>
- <real>0.0</real>
- <key>twopiSeparation</key>
- <real>0.0</real>
- </dict>
- <key>LinksVisible</key>
- <string>NO</string>
- <key>MagnetsVisible</key>
- <string>NO</string>
- <key>MasterSheets</key>
- <array/>
- <key>ModificationDate</key>
- <string>2016-02-09 16:03:50 +0000</string>
- <key>Modifier</key>
- <string>Franck Perignon</string>
- <key>NotesVisible</key>
- <string>NO</string>
- <key>Orientation</key>
- <integer>2</integer>
- <key>OriginVisible</key>
- <string>NO</string>
- <key>PageBreaks</key>
- <string>YES</string>
- <key>PrintInfo</key>
- <dict>
- <key>NSBottomMargin</key>
- <array>
- <string>float</string>
- <string>41</string>
- </array>
- <key>NSHorizonalPagination</key>
- <array>
- <string>coded</string>
- <string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFxlwCG</string>
- </array>
- <key>NSLeftMargin</key>
- <array>
- <string>float</string>
- <string>18</string>
- </array>
- <key>NSPaperSize</key>
- <array>
- <string>size</string>
- <string>{595, 842}</string>
- </array>
- <key>NSPrintReverseOrientation</key>
- <array>
- <string>int</string>
- <string>0</string>
- </array>
- <key>NSRightMargin</key>
- <array>
- <string>float</string>
- <string>18</string>
- </array>
- <key>NSTopMargin</key>
- <array>
- <string>float</string>
- <string>18</string>
- </array>
- </dict>
- <key>PrintOnePage</key>
- <false/>
- <key>ReadOnly</key>
- <string>NO</string>
- <key>RowAlign</key>
- <integer>1</integer>
- <key>RowSpacing</key>
- <real>36</real>
- <key>SheetTitle</key>
- <string>Canevas 1</string>
- <key>SmartAlignmentGuidesActive</key>
- <string>YES</string>
- <key>SmartDistanceGuidesActive</key>
- <string>YES</string>
- <key>UniqueID</key>
- <integer>1</integer>
- <key>UseEntirePage</key>
- <false/>
- <key>VPages</key>
- <integer>1</integer>
- <key>WindowInfo</key>
- <dict>
- <key>CurrentSheet</key>
- <integer>0</integer>
- <key>ExpandedCanvases</key>
- <array/>
- <key>Frame</key>
- <string>{{332, -67}, {946, 873}}</string>
- <key>ListView</key>
- <true/>
- <key>OutlineWidth</key>
- <integer>142</integer>
- <key>RightSidebar</key>
- <false/>
- <key>ShowRuler</key>
- <true/>
- <key>Sidebar</key>
- <true/>
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- 304.723 42.152 304.723 41.738 c 304.723 41.32 305.059 40.984 305.477 40.984
- c 305.891 40.984 306.23 41.32 306.23 41.738 c h
-S Q
-297.605 98.812 m 297.605 98.398 297.27 98.058 296.852 98.058 c 296.434
-98.058 296.098 98.398 296.098 98.812 c 296.098 99.23 296.434 99.566 296.852
- 99.566 c 297.27 99.566 297.605 99.23 297.605 98.812 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-297.605 35.562 m 297.605 35.977 297.27 36.316 296.852 36.316 c 296.434
-36.316 296.098 35.977 296.098 35.562 c 296.098 35.145 296.434 34.809 296.852
- 34.809 c 297.27 34.809 297.605 35.145 297.605 35.562 c h
-S Q
-299.652 78.445 m 299.652 78.031 299.316 77.691 298.898 77.691 c 298.484
- 77.691 298.145 78.031 298.145 78.445 c 298.145 78.863 298.484 79.199 298.898
- 79.199 c 299.316 79.199 299.652 78.863 299.652 78.445 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-299.652 55.93 m 299.652 56.344 299.316 56.684 298.898 56.684 c 298.484
-56.684 298.145 56.344 298.145 55.93 c 298.145 55.512 298.484 55.176 298.898
- 55.176 c 299.316 55.176 299.652 55.512 299.652 55.93 c h
-S Q
-312.379 90.887 m 312.379 90.472 312.043 90.133 311.625 90.133 c 311.207
- 90.133 310.871 90.472 310.871 90.887 c 310.871 91.305 311.207 91.64 311.625
- 91.64 c 312.043 91.64 312.379 91.305 312.379 90.887 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-312.379 43.488 m 312.379 43.902 312.043 44.242 311.625 44.242 c 311.207
- 44.242 310.871 43.902 310.871 43.488 c 310.871 43.07 311.207 42.734 311.625
- 42.734 c 312.043 42.734 312.379 43.07 312.379 43.488 c h
-S Q
-304.789 95.898 m 304.789 95.48 304.453 95.144 304.035 95.144 c 303.617
-95.144 303.281 95.48 303.281 95.898 c 303.281 96.312 303.617 96.652 304.035
- 96.652 c 304.453 96.652 304.789 96.312 304.789 95.898 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-304.789 38.477 m 304.789 38.895 304.453 39.23 304.035 39.23 c 303.617 39.23
- 303.281 38.895 303.281 38.477 c 303.281 38.062 303.617 37.723 304.035 37.723
- c 304.453 37.723 304.789 38.062 304.789 38.477 c h
-S Q
-307.617 79.847 m 307.617 79.43 307.281 79.094 306.863 79.094 c 306.449
-79.094 306.109 79.43 306.109 79.847 c 306.109 80.262 306.449 80.601 306.863
- 80.601 c 307.281 80.601 307.617 80.262 307.617 79.847 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-307.617 54.527 m 307.617 54.945 307.281 55.281 306.863 55.281 c 306.449
- 55.281 306.109 54.945 306.109 54.527 c 306.109 54.113 306.449 53.773 306.863
- 53.773 c 307.281 53.773 307.617 54.113 307.617 54.527 c h
-S Q
-305.086 79.105 m 305.086 78.691 304.746 78.351 304.332 78.351 c 303.914
- 78.351 303.578 78.691 303.578 79.105 c 303.578 79.523 303.914 79.859 304.332
- 79.859 c 304.746 79.859 305.086 79.523 305.086 79.105 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-305.086 55.27 m 305.086 55.684 304.746 56.023 304.332 56.023 c 303.914
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- 54.516 c 304.746 54.516 305.086 54.852 305.086 55.27 c h
-S Q
-271.047 90.535 m 271.047 90.121 270.711 89.781 270.293 89.781 c 269.879
- 89.781 269.539 90.121 269.539 90.535 c 269.539 90.953 269.879 91.289 270.293
- 91.289 c 270.711 91.289 271.047 90.953 271.047 90.535 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-271.047 43.84 m 271.047 44.254 270.711 44.594 270.293 44.594 c 269.879
-44.594 269.539 44.254 269.539 43.84 c 269.539 43.422 269.879 43.086 270.293
- 43.086 c 270.711 43.086 271.047 43.422 271.047 43.84 c h
-S Q
-289.043 95.531 m 289.043 95.117 288.707 94.777 288.289 94.777 c 287.871
- 94.777 287.535 95.117 287.535 95.531 c 287.535 95.949 287.871 96.285 288.289
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-f
-q 1 0 0 -1 0 134.374832 cm
-289.043 38.844 m 289.043 39.258 288.707 39.598 288.289 39.598 c 287.871
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- 38.09 c 288.707 38.09 289.043 38.426 289.043 38.844 c h
-S Q
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- 290.332 98.402 290.332 98.82 c 290.332 99.234 290.672 99.574 291.086 99.574
- c 291.504 99.574 291.84 99.234 291.84 98.82 c h
-f
-q 1 0 0 -1 0 134.374832 cm
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- 290.332 35.973 290.332 35.555 c 290.332 35.141 290.672 34.801 291.086 34.801
- c 291.504 34.801 291.84 35.141 291.84 35.555 c h
-S Q
-273.941 92.433 m 273.941 92.015 273.605 91.68 273.188 91.68 c 272.773 91.68
- 272.434 92.015 272.434 92.433 c 272.434 92.851 272.773 93.187 273.188 93.187
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-f
-q 1 0 0 -1 0 134.374832 cm
-273.941 41.941 m 273.941 42.359 273.605 42.695 273.188 42.695 c 272.773
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- 41.188 c 273.605 41.188 273.941 41.523 273.941 41.941 c h
-S Q
-274.234 97.433 m 274.234 97.015 273.898 96.68 273.48 96.68 c 273.066 96.68
- 272.727 97.015 272.727 97.433 c 272.727 97.847 273.066 98.187 273.48 98.187
- c 273.898 98.187 274.234 97.847 274.234 97.433 c h
-f
-q 1 0 0 -1 0 134.374832 cm
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- 36.188 c 273.898 36.188 274.234 36.527 274.234 36.941 c h
-S Q
-278.746 96.535 m 278.746 96.121 278.41 95.781 277.992 95.781 c 277.578
-95.781 277.238 96.121 277.238 96.535 c 277.238 96.953 277.578 97.289 277.992
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-f
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- 277.238 38.254 277.238 37.84 c 277.238 37.422 277.578 37.086 277.992 37.086
- c 278.41 37.086 278.746 37.422 278.746 37.84 c h
-S Q
-270.734 94.387 m 270.734 93.969 270.398 93.633 269.98 93.633 c 269.562
-93.633 269.227 93.969 269.227 94.387 c 269.227 94.805 269.562 95.14 269.98
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-f
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-270.734 39.988 m 270.734 40.406 270.398 40.742 269.98 40.742 c 269.562
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- 39.234 c 270.398 39.234 270.734 39.57 270.734 39.988 c h
-S Q
-267.77 80.976 m 267.77 80.094 267.055 79.375 266.172 79.375 c 265.285 79.375
- 264.57 80.094 264.57 80.976 c 264.57 81.859 265.285 82.578 266.172 82.578
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- 51.797 267.77 52.516 267.77 53.398 c h
-S Q
-300.578 100.461 m 300.578 100.043 300.242 99.707 299.824 99.707 c 299.41
- 99.707 299.07 100.043 299.07 100.461 c 299.07 100.875 299.41 101.215 299.824
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-f
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- c 300.242 33.16 300.578 33.5 300.578 33.914 c h
-S Q
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- c h
-S Q
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- c h
-S Q
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-S Q
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- c h
-S Q
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- c h
-S Q
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--307.237 -49.138 -307.565 -49.463 -307.567 -49.866 c -307.565 -50.268 -307.24
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- c h
-S Q
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- 104.515 322.961 104.754 323.07 105.156 c 323.176 105.558 322.938 105.972
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-S Q
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- 104.668 328.586 104.906 328.691 105.308 c 328.801 105.711 328.562 106.125
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-S Q
-332.438 111.305 m 332.328 110.902 332.566 110.492 332.969 110.383 c 333.371
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-h
-S Q
-314.773 95.207 m 314.668 94.805 314.906 94.394 315.309 94.285 c 315.711
- 94.18 316.125 94.418 316.23 94.82 c 316.34 95.222 316.098 95.637 315.695
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- c h
-S Q
-316.75 89.469 m 316.75 89.055 316.414 88.715 315.996 88.715 c 315.582 88.715
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-f
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- c 316.414 44.152 316.75 44.488 316.75 44.906 c h
-S Q
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- 318.316 97.418 318.426 97.82 c 318.531 98.222 318.293 98.637 317.891 98.742
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- c h
-S Q
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- 118.926 342.004 119.168 342.113 119.57 c 342.219 119.972 341.98 120.383
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-S Q
-0.803922 0.0196077 0.669358 rg
-202.562 91.164 m 202.562 90.746 202.227 90.41 201.809 90.41 c 201.395 90.41
- 201.055 90.746 201.055 91.164 c 201.055 91.582 201.395 91.918 201.809 91.918
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-f
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-0.376301 w
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- 42.457 c 202.227 42.457 202.562 42.793 202.562 43.211 c h
-S Q
-0.803922 0.0196077 0.669358 rg
-202.883 93.531 m 202.883 93.117 202.543 92.777 202.129 92.777 c 201.711
- 92.777 201.375 93.117 201.375 93.531 c 201.375 93.949 201.711 94.285 202.129
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- 40.09 c 202.543 40.09 202.883 40.426 202.883 40.844 c h
-S Q
-0.803922 0.0196077 0.669358 rg
-207.902 95.336 m 207.902 94.918 207.562 94.582 207.148 94.582 c 206.73
-94.582 206.395 94.918 206.395 95.336 c 206.395 95.754 206.73 96.09 207.148
- 96.09 c 207.562 96.09 207.902 95.754 207.902 95.336 c h
-f
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- 38.285 c 207.562 38.285 207.902 38.621 207.902 39.039 c h
-S Q
-0.803922 0.0196077 0.669358 rg
-201.762 87.808 m 201.762 87.394 201.426 87.055 201.008 87.055 c 200.594
- 87.055 200.254 87.394 200.254 87.808 c 200.254 88.226 200.594 88.566 201.008
- 88.566 c 201.426 88.566 201.762 88.226 201.762 87.808 c h
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- 200.254 46.98 200.254 46.566 c 200.254 46.148 200.594 45.809 201.008 45.809
- c 201.426 45.809 201.762 46.148 201.762 46.566 c h
-S Q
-0.803922 0.0196077 0.669358 rg
-206.383 90.633 m 206.383 90.219 206.043 89.879 205.629 89.879 c 205.211
- 89.879 204.875 90.219 204.875 90.633 c 204.875 91.051 205.211 91.387 205.629
- 91.387 c 206.043 91.387 206.383 91.051 206.383 90.633 c h
-f
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- 44.496 204.875 44.156 204.875 43.742 c 204.875 43.324 205.211 42.988 205.629
- 42.988 c 206.043 42.988 206.383 43.324 206.383 43.742 c h
-S Q
-0.803922 0.0196077 0.669358 rg
-209.742 97.211 m 209.742 96.793 209.402 96.457 208.988 96.457 c 208.57
-96.457 208.23 96.793 208.23 97.211 c 208.23 97.625 208.57 97.965 208.988
- 97.965 c 209.402 97.965 209.742 97.625 209.742 97.211 c h
-f
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-S Q
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-S Q
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- 65.367 27.207 65.336 26.14 c 65.27 23.887 64.457 22.035 62.816 20.379 c
- 62.508 20.066 61.59 19.281 61.543 19.285 c 61.516 19.285 61.41 19.898 61.309
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- 25.289 60.691 26.492 c 60.668 28.472 60.922 29.789 61.551 30.914 c 61.781
- 31.328 62.031 31.668 62.703 32.484 c 63.52 33.476 63.469 33.512 63.898
-34.23 c 64.293 34.89 l h
-f
-145.109 35.664 m 144.395 33.926 145.219 31.074 145.762 28.137 c 145.789
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- 29.453 147.867 28.574 148.078 27.523 c 148.516 25.316 148.145 23.328 146.918
- 21.344 c 146.688 20.972 145.973 20 145.926 19.992 c 145.898 19.992 145.66
- 20.562 145.383 21.293 c 145.262 21.613 145.02 22.246 144.844 22.699 c 144.117
- 24.574 143.762 25.652 143.473 26.824 c 143.004 28.746 142.957 30.086 143.312
- 31.324 c 143.445 31.781 143.609 32.168 144.082 33.113 c 144.656 34.265
-144.598 34.289 144.855 35.086 c 145.09 35.816 l h
-f
-311.117 83.683 m 311.117 83.265 310.777 82.93 310.363 82.93 c 309.945 82.93
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- c 310.777 84.437 311.117 84.097 311.117 83.683 c h
-f
-q 1 0 0 -1 0 134.374832 cm
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- 51.445 309.609 51.109 309.609 50.691 c 309.609 50.277 309.945 49.938 310.363
- 49.938 c 310.777 49.938 311.117 50.277 311.117 50.691 c h
-S Q
-287.637 69.121 m 287.637 68.238 286.922 67.519 286.039 67.519 c 285.152
- 67.519 284.438 68.238 284.438 69.121 c 284.438 70.008 285.152 70.722 286.039
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-f
-0.798797 w
-q 1 0 0 -1 0 134.374832 cm
-287.637 65.254 m 287.637 66.137 286.922 66.855 286.039 66.855 c 285.152
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- 63.652 c 286.922 63.652 287.637 64.367 287.637 65.254 c h
-S Q
-288.438 64.43 m 288.438 63.547 287.723 62.828 286.836 62.828 c 285.953
-62.828 285.238 63.547 285.238 64.43 c 285.238 65.312 285.953 66.031 286.836
- 66.031 c 287.723 66.031 288.438 65.312 288.438 64.43 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-288.438 69.945 m 288.438 70.828 287.723 71.547 286.836 71.547 c 285.953
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- 68.344 c 287.723 68.344 288.438 69.062 288.438 69.945 c h
-S Q
-282.738 68.121 m 282.738 67.238 282.023 66.519 281.137 66.519 c 280.254
- 66.519 279.539 67.238 279.539 68.121 c 279.539 69.008 280.254 69.722 281.137
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-f
-q 1 0 0 -1 0 134.374832 cm
-282.738 66.254 m 282.738 67.137 282.023 67.855 281.137 67.855 c 280.254
- 67.855 279.539 67.137 279.539 66.254 c 279.539 65.367 280.254 64.652 281.137
- 64.652 c 282.023 64.652 282.738 65.367 282.738 66.254 c h
-S Q
-277.539 77.82 m 277.539 76.937 276.82 76.222 275.938 76.222 c 275.055 76.222
- 274.336 76.937 274.336 77.82 c 274.336 78.707 275.055 79.422 275.938 79.422
- c 276.82 79.422 277.539 78.707 277.539 77.82 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-277.539 56.555 m 277.539 57.438 276.82 58.152 275.938 58.152 c 275.055
-58.152 274.336 57.438 274.336 56.555 c 274.336 55.668 275.055 54.953 275.938
- 54.953 c 276.82 54.953 277.539 55.668 277.539 56.555 c h
-S Q
-286.777 56.32 m 286.777 55.086 285.773 54.082 284.539 54.082 c 283.301
-54.082 282.297 55.086 282.297 56.32 c 282.297 57.558 283.301 58.562 284.539
- 58.562 c 285.773 58.562 286.777 57.558 286.777 56.32 c h
-f
-1.118315 w
-q 1 0 0 -1 0 134.374832 cm
-286.777 78.055 m 286.777 79.289 285.773 80.293 284.539 80.293 c 283.301
- 80.293 282.297 79.289 282.297 78.055 c 282.297 76.816 283.301 75.812 284.539
- 75.812 c 285.773 75.812 286.777 76.816 286.777 78.055 c h
-S Q
-283.578 62.422 m 283.578 61.183 282.574 60.18 281.336 60.18 c 280.102 60.18
- 279.098 61.183 279.098 62.422 c 279.098 63.66 280.102 64.664 281.336 64.664
- c 282.574 64.664 283.578 63.66 283.578 62.422 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-283.578 71.953 m 283.578 73.191 282.574 74.195 281.336 74.195 c 280.102
- 74.195 279.098 73.191 279.098 71.953 c 279.098 70.715 280.102 69.711 281.336
- 69.711 c 282.574 69.711 283.578 70.715 283.578 71.953 c h
-S Q
-266.68 74.922 m 266.68 73.683 265.676 72.68 264.438 72.68 c 263.199 72.68
- 262.195 73.683 262.195 74.922 c 262.195 76.16 263.199 77.164 264.438 77.164
- c 265.676 77.164 266.68 76.16 266.68 74.922 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-266.68 59.453 m 266.68 60.691 265.676 61.695 264.438 61.695 c 263.199 61.695
- 262.195 60.691 262.195 59.453 c 262.195 58.215 263.199 57.211 264.438 57.211
- c 265.676 57.211 266.68 58.215 266.68 59.453 c h
-S Q
-276.48 72.422 m 276.48 71.183 275.477 70.18 274.238 70.18 c 273 70.18 271.996
- 71.183 271.996 72.422 c 271.996 73.66 273 74.664 274.238 74.664 c 275.477
- 74.664 276.48 73.66 276.48 72.422 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-276.48 61.953 m 276.48 63.191 275.477 64.195 274.238 64.195 c 273 64.195
- 271.996 63.191 271.996 61.953 c 271.996 60.715 273 59.711 274.238 59.711
- c 275.477 59.711 276.48 60.715 276.48 61.953 c h
-S Q
-272.277 77.023 m 272.277 75.785 271.273 74.781 270.039 74.781 c 268.801
- 74.781 267.797 75.785 267.797 77.023 c 267.797 78.258 268.801 79.262 270.039
- 79.262 c 271.273 79.262 272.277 78.258 272.277 77.023 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-272.277 57.352 m 272.277 58.59 271.273 59.594 270.039 59.594 c 268.801
-59.594 267.797 58.59 267.797 57.352 c 267.797 56.117 268.801 55.113 270.039
- 55.113 c 271.273 55.113 272.277 56.117 272.277 57.352 c h
-S Q
-291.438 72.57 m 291.438 71.687 290.723 70.972 289.836 70.972 c 288.953
-70.972 288.238 71.687 288.238 72.57 c 288.238 73.457 288.953 74.172 289.836
- 74.172 c 290.723 74.172 291.438 73.457 291.438 72.57 c h
-f
-0.798797 w
-q 1 0 0 -1 0 134.374832 cm
-291.438 61.805 m 291.438 62.688 290.723 63.402 289.836 63.402 c 288.953
- 63.402 288.238 62.688 288.238 61.805 c 288.238 60.918 288.953 60.203 289.836
- 60.203 c 290.723 60.203 291.438 60.918 291.438 61.805 c h
-S Q
-292.539 67.07 m 292.539 66.187 291.82 65.472 290.938 65.472 c 290.055 65.472
- 289.336 66.187 289.336 67.07 c 289.336 67.957 290.055 68.672 290.938 68.672
- c 291.82 68.672 292.539 67.957 292.539 67.07 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-292.539 67.305 m 292.539 68.188 291.82 68.902 290.938 68.902 c 290.055
-68.902 289.336 68.188 289.336 67.305 c 289.336 66.418 290.055 65.703 290.938
- 65.703 c 291.82 65.703 292.539 66.418 292.539 67.305 c h
-S Q
-284.738 72.773 m 284.738 71.887 284.023 71.172 283.137 71.172 c 282.254
- 71.172 281.539 71.887 281.539 72.773 c 281.539 73.656 282.254 74.371 283.137
- 74.371 c 284.023 74.371 284.738 73.656 284.738 72.773 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-284.738 61.602 m 284.738 62.488 284.023 63.203 283.137 63.203 c 282.254
- 63.203 281.539 62.488 281.539 61.602 c 281.539 60.719 282.254 60.004 283.137
- 60.004 c 284.023 60.004 284.738 60.719 284.738 61.602 c h
-S Q
-297.438 67.871 m 297.438 66.988 296.723 66.269 295.836 66.269 c 294.953
- 66.269 294.238 66.988 294.238 67.871 c 294.238 68.758 294.953 69.472 295.836
- 69.472 c 296.723 69.472 297.438 68.758 297.438 67.871 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-297.438 66.504 m 297.438 67.387 296.723 68.105 295.836 68.105 c 294.953
- 68.105 294.238 67.387 294.238 66.504 c 294.238 65.617 294.953 64.902 295.836
- 64.902 c 296.723 64.902 297.438 65.617 297.438 66.504 c h
-S Q
-281.238 81.172 m 281.238 80.289 280.523 79.57 279.637 79.57 c 278.754 79.57
- 278.039 80.289 278.039 81.172 c 278.039 82.055 278.754 82.773 279.637 82.773
- c 280.523 82.773 281.238 82.055 281.238 81.172 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-281.238 53.203 m 281.238 54.086 280.523 54.805 279.637 54.805 c 278.754
- 54.805 278.039 54.086 278.039 53.203 c 278.039 52.32 278.754 51.602 279.637
- 51.602 c 280.523 51.602 281.238 52.32 281.238 53.203 c h
-S Q
-280.738 74.172 m 280.738 73.289 280.023 72.57 279.137 72.57 c 278.254 72.57
- 277.539 73.289 277.539 74.172 c 277.539 75.055 278.254 75.773 279.137 75.773
- c 280.023 75.773 280.738 75.055 280.738 74.172 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-280.738 60.203 m 280.738 61.086 280.023 61.805 279.137 61.805 c 278.254
- 61.805 277.539 61.086 277.539 60.203 c 277.539 59.32 278.254 58.602 279.137
- 58.602 c 280.023 58.602 280.738 59.32 280.738 60.203 c h
-S Q
-274.34 81.57 m 274.34 80.687 273.621 79.972 272.738 79.972 c 271.855 79.972
- 271.137 80.687 271.137 81.57 c 271.137 82.457 271.855 83.172 272.738 83.172
- c 273.621 83.172 274.34 82.457 274.34 81.57 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-274.34 52.805 m 274.34 53.688 273.621 54.402 272.738 54.402 c 271.855 54.402
- 271.137 53.688 271.137 52.805 c 271.137 51.918 271.855 51.203 272.738 51.203
- c 273.621 51.203 274.34 51.918 274.34 52.805 c h
-S Q
-287.219 75.797 m 287.219 75.18 286.715 74.676 286.098 74.676 c 285.48 74.676
- 284.977 75.18 284.977 75.797 c 284.977 76.418 285.48 76.918 286.098 76.918
- c 286.715 76.918 287.219 76.418 287.219 75.797 c h
-f
-0.559158 w
-q 1 0 0 -1 0 134.374832 cm
-287.219 58.578 m 287.219 59.195 286.715 59.699 286.098 59.699 c 285.48
-59.699 284.977 59.195 284.977 58.578 c 284.977 57.957 285.48 57.457 286.098
- 57.457 c 286.715 57.457 287.219 57.957 287.219 58.578 c h
-S Q
-283.434 76.965 m 283.434 76.344 282.934 75.844 282.312 75.844 c 281.695
- 75.844 281.191 76.344 281.191 76.965 c 281.191 77.582 281.695 78.086 282.312
- 78.086 c 282.934 78.086 283.434 77.582 283.434 76.965 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-283.434 57.41 m 283.434 58.031 282.934 58.531 282.313 58.531 c 281.695
-58.531 281.191 58.031 281.191 57.41 c 281.191 56.793 281.695 56.289 282.313
- 56.289 c 282.934 56.289 283.434 56.793 283.434 57.41 c h
-S Q
-290.504 77.246 m 290.504 76.629 290.004 76.125 289.383 76.125 c 288.766
- 76.125 288.266 76.629 288.266 77.246 c 288.266 77.867 288.766 78.367 289.383
- 78.367 c 290.004 78.367 290.504 77.867 290.504 77.246 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-290.504 57.129 m 290.504 57.746 290.004 58.25 289.383 58.25 c 288.766 58.25
- 288.266 57.746 288.266 57.129 c 288.266 56.508 288.766 56.008 289.383 56.008
- c 290.004 56.008 290.504 56.508 290.504 57.129 c h
-S Q
-286.547 79.367 m 286.547 78.75 286.043 78.246 285.426 78.246 c 284.805
-78.246 284.305 78.75 284.305 79.367 c 284.305 79.988 284.805 80.488 285.426
- 80.488 c 286.043 80.488 286.547 79.988 286.547 79.367 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-286.547 55.008 m 286.547 55.625 286.043 56.129 285.426 56.129 c 284.805
- 56.129 284.305 55.625 284.305 55.008 c 284.305 54.387 284.805 53.887 285.426
- 53.887 c 286.043 53.887 286.547 54.387 286.547 55.008 c h
-S Q
-295.031 71.73 m 295.031 71.113 294.527 70.609 293.91 70.609 c 293.293 70.609
- 292.789 71.113 292.789 71.73 c 292.789 72.351 293.293 72.851 293.91 72.851
- c 294.527 72.851 295.031 72.351 295.031 71.73 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-295.031 62.645 m 295.031 63.262 294.527 63.766 293.91 63.766 c 293.293
-63.766 292.789 63.262 292.789 62.645 c 292.789 62.023 293.293 61.523 293.91
- 61.523 c 294.527 61.523 295.031 62.023 295.031 62.645 c h
-S Q
-284.281 82.195 m 284.281 81.578 283.781 81.074 283.16 81.074 c 282.543
-81.074 282.043 81.578 282.043 82.195 c 282.043 82.816 282.543 83.316 283.16
- 83.316 c 283.781 83.316 284.281 82.816 284.281 82.195 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-284.281 52.18 m 284.281 52.797 283.781 53.301 283.16 53.301 c 282.543 53.301
- 282.043 52.797 282.043 52.18 c 282.043 51.559 282.543 51.059 283.16 51.059
- c 283.781 51.059 284.281 51.559 284.281 52.18 c h
-S Q
-277.496 83.609 m 277.496 82.992 276.992 82.488 276.375 82.488 c 275.754
- 82.488 275.254 82.992 275.254 83.609 c 275.254 84.23 275.754 84.73 276.375
- 84.73 c 276.992 84.73 277.496 84.23 277.496 83.609 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-277.496 50.766 m 277.496 51.383 276.992 51.887 276.375 51.887 c 275.754
- 51.887 275.254 51.383 275.254 50.766 c 275.254 50.145 275.754 49.645 276.375
- 49.645 c 276.992 49.645 277.496 50.145 277.496 50.766 c h
-S Q
-274.242 86.015 m 274.242 85.394 273.738 84.894 273.121 84.894 c 272.504
- 84.894 272 85.394 272 86.015 c 272 86.633 272.504 87.137 273.121 87.137
- c 273.738 87.137 274.242 86.633 274.242 86.015 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-274.242 48.359 m 274.242 48.98 273.738 49.48 273.121 49.48 c 272.504 49.48
- 272 48.98 272 48.359 c 272 47.742 272.504 47.238 273.121 47.238 c 273.738
- 47.238 274.242 47.742 274.242 48.359 c h
-S Q
-277.496 87.43 m 277.496 86.808 276.992 86.308 276.375 86.308 c 275.754
-86.308 275.254 86.808 275.254 87.43 c 275.254 88.047 275.754 88.551 276.375
- 88.551 c 276.992 88.551 277.496 88.047 277.496 87.43 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-277.496 46.945 m 277.496 47.566 276.992 48.066 276.375 48.066 c 275.754
- 48.066 275.254 47.566 275.254 46.945 c 275.254 46.328 275.754 45.824 276.375
- 45.824 c 276.992 45.824 277.496 46.328 277.496 46.945 c h
-S Q
-281.453 85.59 m 281.453 84.972 280.953 84.469 280.332 84.469 c 279.715
-84.469 279.215 84.972 279.215 85.59 c 279.215 86.211 279.715 86.711 280.332
- 86.711 c 280.953 86.711 281.453 86.211 281.453 85.59 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-281.453 48.785 m 281.453 49.402 280.953 49.906 280.332 49.906 c 279.715
- 49.906 279.215 49.402 279.215 48.785 c 279.215 48.164 279.715 47.664 280.332
- 47.664 c 280.953 47.664 281.453 48.164 281.453 48.785 c h
-S Q
-280.605 88.703 m 280.605 88.082 280.105 87.582 279.484 87.582 c 278.867
- 87.582 278.363 88.082 278.363 88.703 c 278.363 89.32 278.867 89.82 279.484
- 89.82 c 280.105 89.82 280.605 89.32 280.605 88.703 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-280.605 45.672 m 280.605 46.293 280.105 46.793 279.484 46.793 c 278.867
- 46.793 278.363 46.293 278.363 45.672 c 278.363 45.055 278.867 44.555 279.484
- 44.555 c 280.105 44.555 280.605 45.055 280.605 45.672 c h
-S Q
-294.324 75.551 m 294.324 74.93 293.82 74.43 293.203 74.43 c 292.586 74.43
- 292.082 74.93 292.082 75.551 c 292.082 76.168 292.586 76.672 293.203 76.672
- c 293.82 76.672 294.324 76.168 294.324 75.551 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-294.324 58.824 m 294.324 59.445 293.82 59.945 293.203 59.945 c 292.586
-59.945 292.082 59.445 292.082 58.824 c 292.082 58.207 292.586 57.703 293.203
- 57.703 c 293.82 57.703 294.324 58.207 294.324 58.824 c h
-S Q
-298.391 71.375 m 298.391 70.754 297.887 70.254 297.27 70.254 c 296.652
-70.254 296.148 70.754 296.148 71.375 c 296.148 71.992 296.652 72.496 297.27
- 72.496 c 297.887 72.496 298.391 71.992 298.391 71.375 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-298.391 63 m 298.391 63.621 297.887 64.121 297.27 64.121 c 296.652 64.121
- 296.148 63.621 296.148 63 c 296.148 62.383 296.652 61.879 297.27 61.879
- c 297.887 61.879 298.391 62.383 298.391 63 c h
-S Q
-307.191 67.914 m 307.191 67.293 306.691 66.793 306.07 66.793 c 305.453
-66.793 304.953 67.293 304.953 67.914 c 304.953 68.531 305.453 69.035 306.07
- 69.035 c 306.691 69.035 307.191 68.531 307.191 67.914 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-307.191 66.461 m 307.191 67.082 306.691 67.582 306.07 67.582 c 305.453
-67.582 304.953 67.082 304.953 66.461 c 304.953 65.844 305.453 65.34 306.07
- 65.34 c 306.691 65.34 307.191 65.844 307.191 66.461 c h
-S Q
-273.875 89.41 m 273.875 88.992 273.539 88.656 273.121 88.656 c 272.703
-88.656 272.367 88.992 272.367 89.41 c 272.367 89.824 272.703 90.164 273.121
- 90.164 c 273.539 90.164 273.875 89.824 273.875 89.41 c h
-f
-0.376301 w
-q 1 0 0 -1 0 134.374832 cm
-273.875 44.965 m 273.875 45.383 273.539 45.719 273.121 45.719 c 272.703
- 45.719 272.367 45.383 272.367 44.965 c 272.367 44.551 272.703 44.211 273.121
- 44.211 c 273.539 44.211 273.875 44.551 273.875 44.965 c h
-S Q
-292.316 78.773 m 292.316 78.359 291.977 78.019 291.562 78.019 c 291.145
- 78.019 290.805 78.359 290.805 78.773 c 290.805 79.191 291.145 79.527 291.562
- 79.527 c 291.977 79.527 292.316 79.191 292.316 78.773 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-292.316 55.602 m 292.316 56.016 291.977 56.355 291.562 56.355 c 291.145
- 56.355 290.805 56.016 290.805 55.602 c 290.805 55.184 291.145 54.848 291.562
- 54.848 c 291.977 54.848 292.316 55.184 292.316 55.602 c h
-S Q
-296.207 78.055 m 296.207 77.637 295.871 77.301 295.453 77.301 c 295.035
- 77.301 294.699 77.637 294.699 78.055 c 294.699 78.469 295.035 78.808 295.453
- 78.808 c 295.871 78.808 296.207 78.469 296.207 78.055 c h
-f
-q 1 0 0 -1 0 134.374832 cm
-296.207 56.32 m 296.207 56.738 295.871 57.074 295.453 57.074 c 295.035
-57.074 294.699 56.738 294.699 56.32 c 294.699 55.906 295.035 55.566 295.453
- 55.566 c 295.871 55.566 296.207 55.906 296.207 56.32 c h
-S Q
-302.812 81.347 m 302.812 80.93 302.473 80.594 302.059 80.594 c 301.641
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- </array>
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- </array>
- </dict>
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- <string>{{0, 5.5555555020384793}, {1040.4040303817515, 726.26261926648476}}</string>
- <key>Zoom</key>
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- <key>ZoomValues</key>
- <array>
- <array>
- <string>Canevas 1</string>
- <real>0.99000000953674316</real>
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- </array>
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diff --git a/docs/sphinx/hysop.bib b/docs/sphinx/hysop.bib
deleted file mode 100644
index 50ca77c319dae5fbbc86d7762e338fc9cb3cdeca..0000000000000000000000000000000000000000
--- a/docs/sphinx/hysop.bib
+++ /dev/null
@@ -1,107 +0,0 @@
-@article{keck:hal-02982080,
- TITLE = {{Double-diffusive sedimentation at high Schmidt numbers: Semi-Lagrangian simulations}},
- AUTHOR = {Keck, Jean-Baptiste and Cottet, Georges-Henri and Meiburg, Eckart and Mortazavi, Iraj and Picard, Christophe},
- URL = {https://hal.archives-ouvertes.fr/hal-02982080},
- JOURNAL = {{Physical Review Fluids}},
- PUBLISHER = {{American Physical Society}},
- VOLUME = {6},
- NUMBER = {2},
- PAGES = {L022301},
- YEAR = {2021},
- MONTH = Feb,
- DOI = {10.1103/PhysRevFluids.6.L022301},
- PDF = {https://hal.archives-ouvertes.fr/hal-02982080v2/file/paper_rev.pdf},
- HAL_ID = {hal-02982080},
- HAL_VERSION = {v2},
-}
-@article{etancelin:hal-02965451,
- TITLE = {{Improvement of remeshed Lagrangian methods for the simulation of dissolution processes at pore-scale}},
- AUTHOR = {Etancelin, Jean-Matthieu and Moonen, Peter and Poncet, Philippe},
- URL = {https://hal.archives-ouvertes.fr/hal-02965451},
- JOURNAL = {{Advances in Water Resources}},
- PUBLISHER = {{Elsevier}},
- VOLUME = {146},
- PAGES = {103780},
- YEAR = {2020},
- MONTH = Oct,
- DOI = {10.1016/j.advwatres.2020.103780},
- KEYWORDS = {Three-dimensional flows ; Digital Rock Physics ; Reactive flows ; Dissolution process ; Lagrangian methods ; Porous media},
- PDF = {https://hal.archives-ouvertes.fr/hal-02965451/file/10Final-AWR-HAL.pdf},
- HAL_ID = {hal-02965451},
- HAL_VERSION = {v1},
-}
-@phdthesis{keck:tel-02433509,
- TITLE = {{Numerical modelling and High Performance Computing for sediment flows}},
- AUTHOR = {Keck, Jean-Baptiste},
- URL = {https://tel.archives-ouvertes.fr/tel-02433509},
- NUMBER = {2019GREAM067},
- SCHOOL = {{Universit{\'e} Grenoble Alpes}},
- YEAR = {2019},
- MONTH = Dec,
- KEYWORDS = {Sediment transport ; Dual-Way coupling ; Particle methods ; Double diffusivity ; Hpc ; Gpu ; Transport s{\'e}dimentaire ; Couplage bidirectonnel ; M{\'e}thodes particulaires ; Double diffusivit{\'e} ; Hpc ; Gpu},
- TYPE = {Theses},
- PDF = {https://tel.archives-ouvertes.fr/tel-02433509v2/file/KECK_2019_diffusion.pdf},
- HAL_ID = {tel-02433509},
- HAL_VERSION = {v2},
-}
-@article{mimeau:hal-01483400,
- TITLE = {{Passive control of the flow around a hemisphere using porous media}},
- AUTHOR = {Mimeau, Chlo{\'e} and Mortazavi, Iraj and Cottet, Georges-Henri},
- URL = {https://hal.archives-ouvertes.fr/hal-01483400},
- JOURNAL = {{European Journal of Mechanics - B/Fluids}},
- PUBLISHER = {{Elsevier}},
- VOLUME = {65},
- PAGES = {213--226},
- YEAR = {2017},
- MONTH = Sep,
- DOI = {10.1016/j.euromechflu.2017.03.002},
- KEYWORDS = {Brinkman penalization method ; drag reduction ; porous media ; passive flow control ; flow past a hemisphere ; vortex methods},
- PDF = {https://hal.archives-ouvertes.fr/hal-01483400/file/paper_EJM.pdf},
- HAL_ID = {hal-01483400},
- HAL_VERSION = {v1},
-}
-@phdthesis{mimeau:tel-01178939,
- TITLE = {{Conception et mise en oeuvre de m{\'e}thodes vortex hybrides-fronti{\`e}res immerg{\'e}es pour des milieux solides-fluides-poreux. Application au contr{\^o}le passif d'{\'e}coulements.}},
- AUTHOR = {Mimeau, Chlo{\'e}},
- URL = {https://tel.archives-ouvertes.fr/tel-01178939},
- NUMBER = {2015GREAM018},
- SCHOOL = {{Universit{\'e} Grenoble Alpes}},
- YEAR = {2015},
- MONTH = Jul,
- KEYWORDS = {Hybrid vortex method ; Penalization method ; Passive flow control ; Porous media ; Milieux poreux ; Contr{\^o}le passif d'{\'e}coulements ; M{\'e}thode de p{\'e}nalisation ; M{\'e}thode vortex hybride},
- TYPE = {Theses},
- PDF = {https://tel.archives-ouvertes.fr/tel-01178939/file/MIMEAU_2015_archivage.pdf},
- HAL_ID = {tel-01178939},
- HAL_VERSION = {v1},
-}
-@article{cottet:hal-00991150,
- TITLE = {{High order Semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues}},
- AUTHOR = {Cottet, Georges-Henri and Etancelin, Jean-Matthieu and P{\'e}rignon, Franck and Picard, Christophe},
- URL = {https://hal.archives-ouvertes.fr/hal-00991150},
- JOURNAL = {{ESAIM: Mathematical Modelling and Numerical Analysis}},
- PUBLISHER = {{EDP Sciences}},
- VOLUME = {48},
- NUMBER = {4},
- PAGES = {1029-1060},
- YEAR = {2014},
- MONTH = Jul,
- DOI = {10.1051/m2an/2014009},
- KEYWORDS = {Advection equations ; particle methods ; semi-Lagrangian methods ; GPU computing},
- PDF = {https://hal.archives-ouvertes.fr/hal-00991150/file/paper.pdf},
- HAL_ID = {hal-00991150},
- HAL_VERSION = {v1},
-}
-@phdthesis{etancelin:tel-01094645,
- TITLE = {{Couplage de mod{\`e}les, algorithmes multi-{\'e}chelles et calcul hybride}},
- AUTHOR = {Etancelin, Jean-Matthieu},
- URL = {https://tel.archives-ouvertes.fr/tel-01094645},
- NUMBER = {2014GRENM072},
- SCHOOL = {{Universit{\'e} de Grenoble}},
- YEAR = {2014},
- MONTH = Dec,
- KEYWORDS = {Turbulent flows ; Hybrid computing ; Model coupling ; Particle methods ; M{\'e}thodes particulaires ; Couplage de mod{\`e}les ; Calcul hybride ; {\'E}coulements turbulents},
- TYPE = {Theses},
- PDF = {https://tel.archives-ouvertes.fr/tel-01094645v3/file/ETANCELIN_2014_archivage.pdf},
- HAL_ID = {tel-01094645},
- HAL_VERSION = {v3},
-}
diff --git a/docs/sphinx/users_guide/absorption_bc.rst b/docs/sphinx/users_guide/absorption_bc.rst
deleted file mode 100644
index ea668c336f64323333dfddf1eafda20f529d26b0..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/absorption_bc.rst
+++ /dev/null
@@ -1,91 +0,0 @@
-.. _absorption_bc:
-
-
-Smooth solenoidal vorticity absorption at the outlet
-====================================================
-
-For sake of efficiency when solving the diffusion and the Poisson equation with FFTW library, periodic boundary conditions are prescribed at the
-domain walls in each direction. Then, one needs to apply a specific treatment on the flow fields in order to discard, in the downstream region,
-the eddies coming periodically from the outlet. This is done through an outlet absorption of the vorticity field.
-Let us first define the expression of the corrected velocity u ̃ allowing to recover a uniform flow field at the inlet :
-
-.. math::
-
- \tilde u = f(x) u_{periodic} + (1 - f(x))u_{\infty},
-
-where f is the smoothing function defined as follows :
-
-.. math::
-
- f(x) = \begin{cases}
- 1, \text{ if } x < x_b\\
- \frac{tanh(\epsilon(x - x_c)) - tanh(\epsilon(x_e - x_c))}{tanh(\epsilon(x_b - x_c)) - tanh(\epsilon(x_e - x_c))}, \text{ if } x_b < x < x_e \\
- 0 \text{ if } x > x_e
- \end{cases}
-
-
-with :math:`x_c = \frac{x_b + x_e}{2}` (we refer to the figure below for a schematic description of the absorption process).
-
-
-.. figure:: /figures/vorticity_absorption.*
- :scale: 50 %
-
-One can notice that this velocity correction is only performed in the x-direction, that is to say in the
-flow direction: the periodicity is indeed maintained in the y and z directions.
-
-Then, the absorbed vorticity is given by :
-
-.. math::
-
- \omega_{absorbed} &=& \ \nabla \times \tilde u \\
- &=& \ f(x)\omega_{periodic} + \nabla f(x) \times u_{periodic} + \nabla(1 - f(x))\times u_{\infty}\\
- &=&\left[\begin{array}{c}
- \omega_xf(x) \\
- \omega_yf(x) + f'(x)(u_\infty - u_z)\\
- \omega_zf(x) - f'(x)(u_\infty - u_y)\\
- \end{array}\right]
-
-
-Let us mention here that this absorption conserves the solenoidal property of the vorticity field, which is
-not the case when the vorticity is absorbed as follows :
-
-.. math::
-
- \omega_{absorbed} = f(x) \omega_{periodic}
-
-Indeed, defining the absorbed vorticity as the curl of the absorbed velocity to project the vorticity in the space of divergence free vector fields
-(see :ref:`vorticity_projection`). In practice, vorticity absorption based on equation 6 is responsible for reflexion effects and produces spurious oscillations (see drag plots for instance).
-From an algorithmic point of view, the vorticity absorption is applied before solving the Poisson equation.
-
-
-The corresponding operator is :class:AbsorptionBC`, build with the standard parameters of all operators, plus:
-
-* a float, req_flow_rate, to set required value of the flow rate at the outlet.
-* a 'x_range' vector, to set the coordinates delimitating the absorption domain, x_range=[xb, xe]
-* an optional 'filter_func' parameter, list of two python functions, used to set the filter function and its derivative
-
-
-Example::
-
- d3d = Discretization([33, 33, 33])
- dom = Box([1., ] * dim)
- v = Field(dom, name='velo', is_vector=True)
- w = Field(dom, name='vorti', is_vector=True)
- # -- Build and apply absorption operator on w --
- # set flowrate
- flowrate = VariableParameter(data=10.)
- # set layer size
- x_range = [0.9, 1.]
- op = AbsorptionBC(v, w, flowrate, x_range,
- discretization=d3d)
- op.discretize()
- op.setup()
-
- simu = Simulation(nb_iter=100)
- simu.initialize()
-
- op.apply(simu)
-
-
-One may also choose user-defined functions to apply in the absorption area (in place of the tanh-like function shown above).
-See the example in test_absorption_bc.py file.
diff --git a/docs/sphinx/users_guide/adaptive_time_step.rst b/docs/sphinx/users_guide/adaptive_time_step.rst
deleted file mode 100644
index 0887490baa227ca16657cf7efdcffd96b9d2bd58..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/adaptive_time_step.rst
+++ /dev/null
@@ -1,120 +0,0 @@
-.. _adaptive_time_step:
-
-.. currentmodule:: hysop.operator
-
-Adaptive Time Step
-------------------
-
-The class :class:`~adapt_timestep.AdaptiveTimeStep` is used to compute a new time-step
-depending on some predifined criteria.
-
-Several criteria are available, depending on which is the stability condition of interest. All cases
-are listed below.
-
-:math:`u` and :math:`\omega` denote respectively the velocity and vorticity fields, while :math:`x_i` represents the space coordinates, d being the dimension of the space.
-
-Details on the different formulations and criteria can be found in :cite:`Chloe-thesis`.
-
-Stability condition for the advection scheme
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-In particle methods, the non-linear stability of the convection/remeshing scheme is ensured under the following condition on the time step:
-
-.. math::
-
- dt_{adv} \leq\frac{LCFL}{\lVert \nabla u \rVert_\infty}
-
-where LCFL (the Lagrangian CFL) must satisfy :math:`LCFL \leq 1`
-
-Based or derived from this condition, the three following criteria are implemented in HySoP:
-
-* 'gradU' :
-
- .. math::
-
- dt \leq \frac{LCFL}{\max\limits_{0 \leq i < d} \ \left|\frac{\partial u_i}{\partial x_i}\right|}
-
-* 'vort' :
-
- .. math::
-
- dt \leq \frac{LCFL}{\max\limits_{0 \leq i < d} \left|\omega_i\right|}
-
-* 'strain' :
-
- .. math::
-
- dt \leq \frac{LCFL}{\max\limits_{0 \leq i < d} \ \sum\limits_{j=0}^{d-1} \left|\epsilon_{i,j}\right|}
-
- where :math:`\epsilon = 0.5(\nabla u + \nabla u^T)` is the strain tensor.
-
- .. math::
-
- \epsilon_{i,j} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})
-
-Stability condition for the stretching scheme
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-* 'stretch' :
-
- .. math::
-
- dt \leq \frac{c_s}{\max\limits_{0 \leq i < d} \ \sum\limits_{j=0}^{d-1} \left|\frac{\partial u_i}{\partial x_j}\right|}
-
-where :math:`c_s` being a constant depending on the time integration scheme used to discretize the stretching equation. The following table gives the values of C for different time integration schemes :
-
-============= ===== ===========
-Scheme order :math:`c_s`
-============= ===== ===========
-Euler 1 2
-Runge-Kutta 2 2 2
-Runge-Kutta 3 3 2.5127
-Runge-Kutta 4 4 2.7853
-============= ===== ===========
-
-CFL-like stability condition
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-The 'classical' CFL condition.
-
-* 'cfl' :
-
- .. math::
-
- dt \leq \frac{CFL\times h_0}{\max\limits_{0 \leq i < d} \left|u_i\right|}
-
-
- :math:`h_0` is the space step in the first direction.
-
-
-
-When several criteria are chosen, the final time step will be set with the minimum value among
-all criteria.
-
-
-Usage
-+++++
-
-.. code::
-
- # Build and init
- op = AdaptiveTimeStep(velocity, vorticity, simulation, lcfl, cfl, chosen_criteria, time_range, maxdt, ...)
- op.discretize()
- op.setup(rwork)
-
- # Compute criteria AND update simulation time step
- op.apply()
- op.wait()
-
-* lcfl and cfl are optional with lcfl=0.125 and cfl=0.5 as default
-* chosen\_criteria (optional) is the list of the criteria to be applied. For example,
- ['vort', 'gradU']. By default, 'vort' criterion is used.
-* time\_range (optional) may be use to define a range where the operator is applied.
- This range is given in 'iteration number'. For example time\_range = [3, 20] means
- the time step will be updated between iteration number 3 and 20.
-* maxdt (optional, default = 9999) may be used to set a maximum value allowed for the time
- step, whatever the results of criteria computation are.
-* finite differences are used to compute the velocity derivatives which means that for some
- criteria the topology must have ghost points.
-* wait call is very important and compulsory, since time-step may be shared between several
- mpi inter-communicators (tasks).
-* see operators/tests/test_adapative_time_step.py for complete examples on the way to use this operator.
diff --git a/docs/sphinx/users_guide/advection.rst b/docs/sphinx/users_guide/advection.rst
deleted file mode 100644
index 40b1d702946774f76559f32716e82f17473f595f..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/advection.rst
+++ /dev/null
@@ -1,99 +0,0 @@
-.. _advection:
-
-
-Advection operator
-==================
-
-Advection of one or more fields, i.e. find field X which solves
-
-.. math::
-
- \frac{\partial{X}}{\partial{t}} + velocity.\nabla X = 0
-
-for a given velocity field and assuming incompressible flow.
-
-All solvers (except pure python) are able to solve advection in a
-bi-level configuration : velocity is known on a coarser grid than the
-advected field. In that case, the velocity is linearly interpolated
-before move particles. Linear interpolation is a default method and
-the only one implemented for the moment.
-
-
-Scales (Multi-CPU) :
---------------------
-
-Scales library is iterfaced into HySoP (using :class:`~hysop.operator.advection.Advection` operator).
-
-* Fortran routines
-* 3D only
-
-Available solvers:
-^^^^^^^^^^^^^^^^^^
-
-* 'p_O2' : order 4 method, corrected to allow large CFL number,
- untagged particles
-* 'p_O4' : order 4 method, corrected to allow large CFL number,
- untagged particles
-* 'p_L2' : limited and corrected :math:`\Lambda_2`
-* 'p_M4' : :math:`\Lambda_{2,1} (=M'_4)` 4 point formula
-* 'p_M6' (default) : :math:`\Lambda_{4,2} (=M'_6)` 6 point formula
-* 'p_M8' : :math:`M8prime` formula
-* 'p_44' : :math:`\Lambda_{4,4}` formula
-* 'p_64' : :math:`\Lambda_{6,4}` formula
-* 'p_66' : :math:`\Lambda_{6,6}` formula
-* 'p_84' : :math:`\Lambda_{8,4}` formula
-
-
-Time integration:
-^^^^^^^^^^^^^^^^^
-
-* Runge-Kutta 2nd order
-
-
-Splitting
-^^^^^^^^^
-
-Computations are performed with a dimensional splitting as follows:
-
-* 'strang' (2nd order):
-
- * X-dir, half time step
- * Y-dir, half time step
- * Z-dir, full time step
- * Y-dir, half time step
- * X-dir, half time step
-
-* 'classic' (1st order):
-
- * X-dir, full time step
- * Y-dir, full time step
- * Z-dir, full time step
-
-
-HySoP (Multi-CPU an Multi-GPU) :
---------------------------------
-
-* 1D, 2D or 3D advection
-* Pure python code
-* Generated OpenCL code
-* Ghosts layers are computed from a maximum CFL number
-
-
-Available solvers:
-^^^^^^^^^^^^^^^^^^
-
-* All :math:`\Lambda_{p,q}` kernels are computed
-
-Time integration:
-^^^^^^^^^^^^^^^^^
-
-Explicit time intergration solvers:
-
-* Euler
-* Runge-Kutta (orders 1, 2, 3, 4, 4_38)
-
-
-Splitting
-^^^^^^^^^
-
-Splitting is handled as a higer level from HySoP :class:`~hysop.numerics.splitting.directional_splitting.DirectionalSplitting` operators.
diff --git a/docs/sphinx/users_guide/biblio.rst b/docs/sphinx/users_guide/biblio.rst
deleted file mode 100644
index 78b7aeff1932166392619898db9b474617843e15..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/biblio.rst
+++ /dev/null
@@ -1,8 +0,0 @@
-.. _biblio:
-
-Bibliography
-============
-
-.. bibliography:: ../hysop.bib
- :cited:
- :style: plain
diff --git a/docs/sphinx/users_guide/custom.rst b/docs/sphinx/users_guide/custom.rst
deleted file mode 100644
index 1fc626afe5c0f4ab63233714f36a236f86836f9f..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/custom.rst
+++ /dev/null
@@ -1,41 +0,0 @@
-.. _custom:
-
-User-defined operator
-=====================
-
-A specific operator class, :class:`~hysop.operator.custom.Custom` is available in hysop to allow users to
-define their own operator.apply, thanks to a python function.
-
-
-.. code::
-
- # v1, v2, v3 some continuous fields
- iop = IOParams('somefile.dat', fileformat=IO.ASCII)
- my_op = Custom(function=my_function, in_fields=[v1, v2], out_fields=[v3],
- diagnostics_shape=(1,2), io_params=iop, ...)
- my_op.discretize()
- my_op.setup()
-
- simu = Simulation(nb_iter=10)
- my_op.apply(simu)
-
-
-A call to my_op.apply(simu) will run 'my_func(simu, [v1, v2], [v3], buffer)'
-and write 'buffer' into a file (defined as usual with iop).
-
-* my_func arguments list MUST match the following::
-
- def my_func(sim, in_fields, out_fields=None, diagnostics=None):
- # optionally compute out_fields depending on in_fields
- out_fields[0].data[1] = 2 * in_fields[0].data[0]
- # optionally compute some diagnostics
- diagnostics[0,0] = sim.time
- diagnostics[0,1] = out_fields[0].data[0].max()
-
-* diagnostics_shape arg must corresponds to the shape of the buffer written in the function
- and buffer must be a 2D array (i.e. diagnostics_shape a tuple like (nx, ny)).
-
-* if diagnostics_shape is None, no file output will be done and obviously in that case,
- diagnostics must not be filled in my_func
-
-* as usual real use-case are available in operator/test/test_custom.py
diff --git a/docs/sphinx/users_guide/differential_operations.rst b/docs/sphinx/users_guide/differential_operations.rst
deleted file mode 100644
index 94552b4ce9e56a7f8c5af47e4b9db48a2a976478..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/differential_operations.rst
+++ /dev/null
@@ -1,8 +0,0 @@
-.. _differential_operations:
-
-.. currentmodule:: hysop.numerics.differential_operations
-
-Vector calculous based on finite differences
---------------------------------------------
-
-To be done ...
diff --git a/docs/sphinx/users_guide/diffusion.rst b/docs/sphinx/users_guide/diffusion.rst
deleted file mode 100644
index 9d2cde362538e2512d1e97956f06d69ee608cd24..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/diffusion.rst
+++ /dev/null
@@ -1,43 +0,0 @@
-.. _diffusion:
-.. currentmodule:: hysop.operator.diffusion
-
-Diffusion Operator
-------------------
-
-
-The class :class:`Diffusion` is used to model and solve the following problem :
-
-.. math::
-
- \frac{\partial \omega}{\partial t} &=& \nu\Delta\omega
-
-for the unknown vector field :math:`\omega` and a given constant viscosity value :math:`\nu`,
-
-with
-
-.. math::
- \omega = \nabla \times velocity \\
- \nabla . velocity = 0
-
-
-
-Examples
-^^^^^^^^
-
-
-::
-
- from hysop.operator.diffusion import Diffusion
- dom = Box()
- d3d = Discretization([33, 33, 33])
- topo = dom.create_topology(d3d)
- om = Field(name='vorticity', is_vector=True, domain=dom)
- om.randomize(topo)
- op = Diffusion(vorticity=om, viscosity=0.3, discretization=topo)
- op.discretize()
- op.setup()
- sim = Simulation(nb_iter=10)
- sim.initialize()
- while not sim.is_over:
- op.apply(sim)
- sim.advance()
diff --git a/docs/sphinx/users_guide/domains.rst b/docs/sphinx/users_guide/domains.rst
deleted file mode 100644
index 89e9a4a1576a21a3f64f1a691cc910fb8af9c902..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/domains.rst
+++ /dev/null
@@ -1,353 +0,0 @@
- .. _domains:
-
-.. currentmodule:: hysop.domain
-
-Domains
-=======
-
-A :class:`~domain.Domain` is the physical space where fields and operators will be defined, associated with a coordinate system.
-A domain is defined with its dimension (1, 2 or 3 dimensions domains are allowed in HySoP), some geometrical properties and some boundary conditions. Each point in the domain is identified with its space coordinates, denoted as :math:`x, y, z`.
-
-At the time only box-shaped domains are available in HySoP, thanks to class :class:`~box.Box`::
-
- from hysop import Box
- # 1D
- b = Box(length=[2.], origin=[-1.])
- # 3D
- dom = Box(length=[1., 3., 5.], origin=[-1., 2., 3])
- # Default
- dom = Box()
- print dom.length
- # coordinates of origin and 'highest' point
- print dom.origin, dom.end
-
-Origin denotes the coordinates of the 'lowest' point in the box.
-Default dimension is 3d, default sides length is 1. in each direction, and default origin position is 0 for each direction::
-
- dom = Box()
-
-Boundary conditions are set to periodic by default, but other types
-are available through :py:attr:`~constants.BoxBoundaryCondition`.
-
-
-Parallelisation of the simulation
-=================================
-
-The simulation can be parallelised in two different manners:
-
-* by tasks, i.e. some 'tasks' are defined and a group of processes will be affected to each task. Each group is autonomous and work in an asynchronous way.
-* by splitting the data : the domain, and the fields defined on this domain, is splitted into sub-domains, each sub-domain is attributed to a process. Then each process works on its own subset of the data, in an asynchronous manner.
-
-Both ways can be combined. Synchronisation is possible and processes or groups can communicate. This is detailed in the following parts.
-
-.. _topologies:
-
-Communicators and tasks
------------------------
-
-Please check :ref:`mpi_utils` for communicator, tasks and other mpi-related definitions.
-
-When created, a domain is automatically associated with a communicator and the processes of the communicator
-with a task. Default task is :obj:`~hysop.constants.DEFAULT_TASK_ID` for all processes, and default communicator
-is main_comm::
-
- from hysop import Box
- from hysop.core.mpi import main_comm, main_size
- from hysop.constants import HYSOP_DEFAULT_TASK_ID
-
- dom = Box()
- assert dom.comm_task == main_comm
- assert dom.tasks_list() == [HYSOP_DEFAULT_TASK_ID, ] * main_size
-
-A process can be affected to one and only one task and dom.comm_task is the communicator associated to the task
-of the current process.
-
-Ok, let us now assume that we need to define three different tasks, in a simulation run with 8 mpi processes.
-The idea is to bind processes 0, 4, 5 and 6 to task 'red', 1, 2, 3 to task 'green' and 7 to task 'yellow', as
-shown in the figure below:
-
-.. image:: /figures/comm_split.*
-
-To do so, one need to set the optional argument 'proc_tasks' of domain, with a list which maps
-processes rank to task id. Indeed, proc_tasks[4] = 12 means that process of rank 4 (in main_comm!)
-is attached to task number 12.
-
-Try the following program with 8 processes to check the result::
-
- from hysop import Box
- from hysop.core.mpi import main_comm, main_size, main_rank
- from hysop.constants import HYSOP_DEFAULT_TASK_ID
-
- RED = 4
- GREEN = 1
- YELLOW = 12
-
- proc_tasks = [RED, ] * main_size
- proc_tasks[1:4] = [GREEN, ] * 3
- proc_tasks[7] = YELLOW
- dom = Box(proc_tasks=proc_tasks)
-
- assert dom.comm_task != main_comm
- print 'process of rank ', main_rank, ' with task ', dom.current_task()
- print 'rank/main_comm = ', main_rank, ' rank/comm_task', dom.comm_task.Get_rank()
-
-Important remarks:
-
-* `dom.comm_task` defines a different object depending on which process you are.
-* the rank of a process in main_comm may be different from its rank in comm_task
-
-Some useful methods:
-
-* `dom.current_task()` : returns the task id of the current process
-* `dom.tasks_in_proc(i)` : returns task id of process number i
-* `dom.is_on_task(tid)` : returns true if the current process belongs to task tid
-
-MPI topologies
---------------
-
-.. currentmodule:: hysop.core.mpi
-
-Domains may be distributed among several mpi processes, to allow parallel process of the simulation.
-MPI cartesian topologies are used to handle the mapping between hysop objects and mpi processes.
-For details about MPI implementation in hysop, check :ref:`mpi_utils`.
-
-The domain is splitted into N parts, N being the number of processes involved in the current task. MPI processes are arranged in topological patterns, on a one-, two- or three-dimensional grid.
-Each process is identified with its rank in the topology and with its coordinates in the grid. The grid dimension may be any number between 1 and the dimension of the domain. The figure below shows an example with 4 mpi processes distributed in a 2-dimensional grid. The physical domain is on the left and each process on the right owns its proper sub-domain. For example process of rank 2 (P2) handles the subdomain C, and has coordinates (1,0) in the grid.
-
-.. image:: /figures/decomp_domain.*
-
-Moreover, each process knows its neighbours in all directions.
-
-
-Basic usage
-^^^^^^^^^^^
-
-:class:`topology.CartesianTopology` objects are used to described this mpi grid layout (we'll see later that it also handles the local space discretisation, i.e. some meshes local to each process).
-Create a new mpi process distribution is quite simple in HySoP and must be managed through the domain on which
-the topology will be defined, thanks to the method :meth:`~hysop.domain.domain.Domain.create_topology`.
-By default mpi can find the 'best' processes distribution, depending on how many processes are available, on
-the dimension of the space and on the way data are saved in memory (C or Fortran order indeed). So the creation routine may be called very simply::
-
- # 2d domain
- dom = Box(length=[1., 1.])
- d2d = Discretization([65, 65])
- # 2d grid of MPI processes
- topo = dom.create_topology(d2d)
-
-Such a code executed with 4 mpi processes will create a topology similar to the one describe on the figure above.
-Notice the 'd2d' argument, used to specified the space discretisation, since, as said before, :class:`topology.Cartesian` are also used to managed the data distribution on some local meshes. Just ignore this d2d for the moment, we will come back to this in the next part.
-
-The standard methods of a topology are::
-
- # return the number of process in the topology
- print topo.size
- # rank of the current process in the topology
- print topo.rank
- # rank of the neighbours of the current process
- # in direction d
- d = 2
- print topo.neighbours[:, d]
- # id of the task owning this topology
- print topo.task_id()
-
-Advanced usage
-^^^^^^^^^^^^^^
-
-In addition to d2d, different arguments are available to create any kind of topology, according to your needs. All of them are detailed in the example below::
-
- # 3d domain
- dom = Box()
- d3d = Discretisation([65, 65, 65])
- # let mpi choose the best distribution
- topo_0 = dom.create_topology(d3d)
- # choose the dimension of the topology. dim = 1, 2 or 3
- topo_1 = dom.create_topology(d3d, dim=2)
- # choose the shape, i.e how many mpi process in each direction
- topo_2 = dom.create_topology(d3d, shape=[3, 1, 2])
- # choose which direction must be distributed
- topo_3 = dom.create_topology(d3d, cutdir=[True, False, True])
-
-*Remarks:*
-
-* shape, cutdir, periodicity: you must specify a value for each direction of the space, whatever is the dimension of the required topology. For example, topo_2 will be a 2d mpi topology but shape length is 3.
- For shape, the number of process in the mpi grid must corresponds to the number of processes in dom.comm_task. The example above must be executed with 6 processes.
-* shape, cutdir and dim are exclusive : choose one and only one among them.
-
-By default, topologies are periodic in each direction. Use isperiodic argument to customize this::
-
- dom = Box(dimension=2)
- d2d = Discretization([33,33])
- #
- topo_4 = dom.create_topology(d2d, shape=[3, 2], isperiodic=[False, False])
- topo_5 = dom.create_topology(d2d, shape=[3, 2], isperiodic=[False, True])
-
-When executed with 6 mpi processes, topo_4 corresponds to left grid and topo_5 to right grid below
-
-.. image:: /figures/periodic_topo.*
-
-Periodic topologies may be useful to work with periodic domains or data.
-
-
-Keep in mind that each process has its own memory space and must use 'message passing' to access to other processes data.
-
-Use a predifined mpi communicator
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-In some cases, it may be necessary to use a previously built mpi communicator (from another soft, from fortran subroutines ...)::
-
- dom = Box(dimension=2)
- d2d = Discretization([33,33])
- topo_4 = dom.create_topology(d2d, cartesian_topology=another_comm)
- # another_comm being previously defined
-
-
-Space discretization
-====================
-
-Since domain and data may be distributed among mpi processes, we must distinguish two 'levels' of discretization:
-
-* **global** : to set the resolution of a mesh on the whole physical domain; done thanks to :class:`~hysop.tools.parameters.Discretization`. This does not depend on the number of mpi processes. This global resolution is just a 'modelisation' parameter. No data or array will be allocated using this resolution. Everything concerning memory is done at process level.
-* **local** : all the parameters of the meshes on a given mpi process, i.e. the discretization of a sub-domain.
-
-To summarize, you must choose a global discretisation for each topology, for which some local discretisations will be computed, depending on the number of processes and on the boundary conditions.
-
-Global discretisation
----------------------
-
-For the moment, only regular grids are available in HySop, defined with the class :class:`~hysop.tools.parameters.Discretization`::
-
- from hysop import Discretization
- d2d = Discretization([65, 29])
- d3d = Discretization([129, 33, 257])
- d3dg = Discretization([129, 33, 257], ghosts=[1, 2, 1])
-
-The two arguments are the resolution of the grid, i.e the number of points (not cells!) in each direction, and the number of points
-in the ghost layer in each direction. Resolution argument is mandatory, while ghosts is set to zero by default.
-This is a 'global' description of the required discretization. What really happens in memory for fields discretize using these objects strongly depends on the number of mpi processes used for the simulation and on the boundary condition types. This will be discussed later.
-
-Local grids
------------
-
-When creating a topology, a global discretization parameter must be provided.
-From this discretization and the topologies parameters (mpi grid layout), local
-grids are computed and saved in the attribute 'mesh' of the topology::
-
- from hysop import Box, Discretization
- d2d = Discretization([65, 65])
- dom = Box(dimension=2)
- topo = dom.create_topology(d2d)
- print topo.mesh
-
-
-To compute the local resolution, the global resolution in each direction is divided by the number of mpi process in this direction. Remaining points are appended on the last process of the considered direction. Then, if asked, some ghost points are added on local boundaries. Definition and utility of ghost points is detailed thereafter.
-
-To clarify notations, an example is detailed in the figure below. A periodic 1d domain of length Lx, starting at point x0 is condisered. The chosen discretization has 11 points, as shown on the top part of the figure. The points are numbered from 0 to 10, which corresponds to their **global index**. Creating a topology based on this discretization, with 2 points in the ghost layer and executed on 3 mpi processes results in the local meshes at the bottom of the figure. Pn stands for process of rank n in the 1d topology.
-
-.. image:: /figures/data_discr.*
-
-We distinguish **computational** points, in black on the figure, from ghost points, in red. The former are 'real' grid points where computation will occur. They corresponds to the points of the original (global) grid. Ghost points are some extra points, some kind of fake boundaries, local to each process.
-Points are identified thanks to their index, either local (numbers in green) or relative to the global mesh (numbers in black).
-
-Important remark:
-
-* when the domain is periodic, the last point in each direction is ignored on local meshes, to save memory. Either this point is in the ghost layer or not present at all if ghost layer equal to 0. Mind this to choose the values of the discretization parameter. A common usage consists in choosing a number of points which is a power of two, plus 1, to fit with fftw requirements.
-
-To conclude, here is a list of the most useful attributes of mesh class::
-
- from hysop import Box, Discretization
- d1d = Discretization([11], ghosts=[2])
- dom = Box(dimension=1)
- topo = dom.create_topology(d1d)
- mesh = topo.mesh
- # local resolution
- print mesh.resolution
- # local resolution excluding ghost points
- print mesh.compute_resolution
- # coordinates of the origin of the local mesh
- print mesh.origin
- # list of local indices of computational points
- print mesh.compute_index
- # list of local indices of all points
- print mesh.local_index
- # coordinates of local points
- print mesh.coords
- # coordinates of computational points
- print mesh.compute_coords
- # global index of the first point of the local mesh
- print mesh.position
- # space step
- print mesh.space_step
-
-
-compute_index returns a list of python slices. For example, in our example of the figure above, on process 1, compute_index is equal to [slice(2, 5)], which means that first point has local index 2 and last point local index 4. This argument can be used to call a numpy array::
-
- import hysop.tools.numpywrapper as npw
- # init some arrays.
- a = npw.zeros(topo.mesh.resolution)
- # set value only for computational points:
- a[mesh.compute_index] = 3.
-
-
-coords or compute_coords returns a tuple, with coordinates values in each direction. It can be used as argument in a function, like this::
-
- from hysop import Box, Discretization
- import numpy as np
-
- def func(x, y):
- return np.cos(x) + np.sin(y)
-
-
- d2d = Discretization([11, 11], ghosts=[2, 2])
- dom = Box(dimension=2)
- topo = dom.create_topology(d2d)
-
- # apply 'func' for (x,y) of each computational point of topo
- # on the current process
- res = func(*topo.mesh.compute_coords)
- # res is an array of shape topo.mesh.compute_resolution
-
-Ghost points
-^^^^^^^^^^^^
-
-Ghost-points are a very common trick to manage the behavior of numerical methods at the boundaries of local (distributed) subdomains.
-
-Consider the left-hand figure below, where for some numerical method with a 5 points stencil, one need to compute some value at point of index 4. To do so, values of two neighboring points in each direction are required.
-If the grid is distributed without ghost points, values from points of global indices 6 and 7 will be held by process 2 while computation for point 5 occurs on process 1. So, some 'false' boundaries are added, the ghost points, in red on the figure. That means that values of points 6 and 7 are duplicated on process 1 and 2. On the former, this points are ghost points, while on process 2 they are computational points.
-
-.. image:: /figures/ghost_points.*
-
-For a time step, the algorithm will then be more or less:
-
-* Update ghost points : each process receive values from its neighbours in its ghost points and send values to ghost points of its neighbours.
-* Compute : each process apply numerical scheme for all its computational points.
-
-When required, the update step is handled by the operator and so hidden from the end user.
-But, if needed, use :class:`~hysop.numerics.update_ghosts.UpdateGhosts` to explicitely update
-ghost points::
-
- from hysop import Box, Discretization
- from hysop.numerics.update_ghosts import UpdateGhost
- import hysop.tools.numpywrapper as npw
- box = Box(dimension=1)
- topo = box.create_topology(Discretization([65], [2]))
- # number of numpy arrays to be updated
- nb = 2
- # create the 'updater' for two arrays
- synchronize = UpdateGhost(topo, nb)
-
- # init some arrays.
- a = npw.zeros(topo.mesh.resolution)
- b = npw.zeros(topo.mesh.resolution)
- a[...] = topo.rank
- b[...] = topo.rank * 3.
-
- # update ghost points
- synchronize.apply([a, b])
-
-A few remarks:
-
-* the number of numpy arrays to be synchronized must be set at init and must corresponds to the length of
- the list argument of the apply method,
-* arrays shapes must fit with the topology local resolution,
-* ghost points implies mpi communications and impact the memory print of the numerical method. Therefore
- they must be used only when required.
diff --git a/docs/sphinx/users_guide/energy_enstrophy.rst b/docs/sphinx/users_guide/energy_enstrophy.rst
deleted file mode 100644
index 8102f437f915fada3686595fc80e06e600bc7310..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/energy_enstrophy.rst
+++ /dev/null
@@ -1,49 +0,0 @@
-.. _energy_enstrophy:
-
-
-Kinetic energy and enstrophy computation
-========================================
-
-
-For :math:`v(x, t), \omega(x,t)` some velocity and vorticity fields, the kinetic energy is defined as :
-
-.. math::
-
- E_c = 0.5V_{\Omega}\int_\Omega v^2 d\Omega
-
-and the enstrophy as :
-
-.. math::
-
- S = V_{\Omega}\int_\Omega \omega^2 d\Omega
-
-
-* Default value for :math:`V_{\Omega}` is :math:`\frac{1.}{||\Omega||}` with :math:`||\Omega||` the volume of the domain of definition of the fields, but it can be set to 1. by setting is_normalized parameter to False in the operator definition.
-* By default, the output file name is energy_enstrophy.
-
-Energy and enstrophy can be computed thanks to :class:`hysop.operator.energy_enstrophy.EnergyEnstrophy` operator
-as shown in the example below.
-
-
-::
-
- from hysop.operators import EnergyEnstrophy
- from hysop import Field, Box, Simulation, Discretization
- d3d = Discretization([65, 65, 65])
- box = Box()
- velocity = Field(domain=box, name='velo', is_vector=True)
- vorticity = Field(domain=box, name='vorti', is_vector=True)
- op = EnergyEnstrophy(velocity, vorticity, discretization=d3d)
- # Or
- # op = EnergyEnstrophy(velocity, vorticity, discretization=d3d
- # is_normalized=False)
- op.discretize()
- op.setup()
- simu = Simulation(nb_iter=100)
- # do some stuff on velocity and vorticity ...
-
- op.apply(simu)
- # --> compute energy and enstrophy for velocity and vorticity
- # values at time simu.time
- print op.energy
- print op.enstrophy
diff --git a/docs/sphinx/users_guide/fields.rst b/docs/sphinx/users_guide/fields.rst
deleted file mode 100644
index 8f63c8477dee947afb18c3bc728c20a2bd058022..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/fields.rst
+++ /dev/null
@@ -1,365 +0,0 @@
- .. _fields:
-
-Continuous and discrete fields
-==============================
-
-.. currentmodule:: hysop.fields
-
-
-.. contents::
-
-Definition
-----------
-
-In HySoP, a **continuous field** (:class:`~hysop.fields.continuous.Field`) is an abstract object which represents the usual vector field, in mathematical sense, i.e some function which associates a scalar or a vector to each point of the space.
-Such objects are used as input and/or output variables for continuous operators and must be defined with at least:
-
-* a :class:`~hysop.domain.domain.Domain` : the physical domain of definition of the field,
-* a name, compulsory (required for i/o).
-
-Scalar fields are built by default. Use is_vector or nb_component optional arguments to build
-n-components fields (n=domain.dimension for vector fields)::
-
- from hysop import Field, Box
-
- # build a domain
- box = Box()
- # Scalar field
- scal = Field(domain=box, name='s1')
- # Vector field
- vec = Field(domain=box, name='v1', is_vector=True)
- # n-component field
- n = 4
- vec_n = Field(domain=box, name='vn', nb_components=n)
- # nb_component = domain.dimension is equivalent to is_vector=True
-
-Obviously one needs to associate one or more topologies (:class:`hysop.mpi.topology.Cartesian`, see :ref:`topologies`) to such a field, to be able to apply operators or numerical methods. Remind that 'topologies' means either the definition of a global space resolution and of the way data are distributed through mpi processes::
-
- # The continuous field
- vec = Field(domain=box, name='vn', is_vector=True)
- # space discretisation and data distribution description
- d3d = Discretization([65, 65, 65])
- topo = box.create_topology(d3d)
- # build a discretisation of vec on d3d and distribute its data according to 'topo'
- vd = vec.discretize(topo)
- vd is vec.discreteFields[topo]
- # --> returns True
-
-Each time 'discretize' function is called, a new entry is added to the continuous field attribute named discreteFields, which maps a topology to a :class:`~hysop.fields.discrete.DiscreteField`. Therefore, a discrete field will handle local arrays of values
-of the field on the local mesh.
-
-To summarize, a **discrete field** (:class:`~hysop.fields.discrete.DiscreteField) is the association of:
-
-* a topology (:class:`hysop.mpi.topology.Cartesian`~),
-* a list of numpy arrays (one per component), values of the field on the local mesh.
-
-For example::
-
- # The continuous field --> no memory allocation
- vec = Field(domain=box, name='vn', is_vector=True)
- # space discretisation and data distribution description
- d3d = Discretization([65, 65, 65])
- topo = box.create_topology(d3d)
- d3d_with_ghosts = Discretization([65, 33, 65], [2, 2, 2])
- topo_g = box.create_topology(d3d_with_ghosts)
- # build a discretisation of vec on d3d and distribute its data according to 'topo'
- vd = vec.discretize(topo)
- vd is vec.discreteFields[topo]
- # --> returns True
- # another data distribution
- vd2 = vec.discretize(topo_g)
-
- for i in range(vec.nb_components):
- print vd.data[i].shape
- print vd2.data[i].shape
-
-will display the local resolution of each component of the discretisations 'topo' and 'topo_g' of vec.
-Their values will depend either on the global resolution, the number of mpi processes on which the program run and
-on the current process rank.
-
-
-Notice that the exact behavior of::
-
- vd = vec.discretize(topo)
-
-is
-
-* check if a discretisation of vec on topo already exists. If so, returns this discretisation (i.e. vec.discreteFields[topo]),
-* else discretizes vec on topo and returns vec.discreteFields[topo].
-
-
-
-**Remark:**
-
-Usually, in a 'complete' problem, the user does not have
-to bother with the topologies' stuff and the fields discretisations.
-All this is hidden from user : one just has to define
-some operators and the fields they work with, how those fields are initialized (see below)
-, bricks of the global problem, associated with
-some required global resolutions and some solving methods.
-Then the setup process of the problem/operators will automatically
-performs the topologies creation and the discretisation of the fields.
-See operators/problem documentation for more details or
-the examples in HySoP/Examples.
-
-
-How to initialize a field?
---------------------------
-
-Either fields values are set to zero (default behavior) or computed with a user-defined function.
-To do so, continuous field has an 'initialize' method. It's behavior depends on the way the field
-has been defined and on the arguments of 'initialize'.
-
-
-*Notice that you can find examples of all the different ways to initialize fields in fields/tests/test_field.py.*
-
-If no function has been associated with the field
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-::
-
- # case 1: set all components values of all discrete fields to zero.
- vec.initialize()
-
- # case 2: set all components values of vec.discreteFields[some_topo] to zero
- vec.initialize(time=0., topo=some_topo)
-
-Remarks:
-
-* time argument in case 2 is not used but must be provided.
-* in case 2, a call to initialize will proceed with the discretisation of vec on some_topo
- if it does not already exist.
-
-Set a function to compute field values
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-A user-defined function can be connected to a field::
-
- # during definition
- vec = Field(domain=box, name='vn', is_vector=True, formula=myfunction)
- # or later
- vec2 = Field(domain=box, name='v2', is_vector=True)
- vec2.set_formula(myfunction, vectorize_formula=True)
-
-myfunction must be a function of time and space, returning n numpy arrays, n being the
-number of components of the field. The exact possible signatures of this function will
-be shown later.
-
-Therefore, to initialize vec, run::
-
- # case 1: use myfunction to compute all components values of all discrete fields
- # at time t=0.
- vec.initialize()
- # case 2: use myfunction to compute all components values of all discrete fields
- # at time t
- t = 2.4
- vec.initialize(time=t)
- # case 3: use myfunction to compute all components values of vec.discreteFields[some_topo]
- # at time t
- vec.initialize(time=t, topo=some_topo)
-
-
-User-defined function must be::
-
- # case 1: 'vectorize' case
- def myfunction(x, y, z, t):
- f_x = ...
- f_y = ...
- f_z = ...
- return f_x, f_y, f_z
-
-
- # case 2:
-
- def myfunction(res, x, y, z, t):
- res[0] = ...
- res[1] = ...
- res[2] = ...
- return res
-
-
-Remarks:
-
-* in 2d, z argument is removed, in 1d, only x remains (with t).
-* Res must be a list of numpy arrays of length equal to the number of
- component of the field to be initialized.
-* The number of returned values in case 1 must be equal to the number of
- component of the field to be initialized.
-* Case1 function takes scalar arguments as inputs and so will be vectorized
- during initialization (about vectorization in numpy, see http://docs.scipy.org/doc/numpy-1.10.1/reference/generated/numpy.vectorize.html), while case2 function works on numpy arrays. Case 2 must be prefered when possible for performance reasons.
-* When vectorization is needed, vectorize_formula=True must be set during definition of the field.
-
-
-Here is an example where a vector field 'velo' on a 2d domain is initialized with a vectorized function
-such that
-
-.. math::
-
- velo_x(x,y,t) &=& cos(x)\\
- velo_y(x,y,t) &=& cos(y) + tx
-
-and a vector field :math:`\omega` on a 3d domain with a 'numpy-ready' function. Notice that this last case is what we call 'Taylor-Green' in many of our examples.
-
-.. math::
-
- \omega_x(x, y, z, t) = - cos(x)sin(y)sin(z)\\
- \omega_y(x, y, z, t) = - sin(x)cos(y)sin(z)\\
- \omega_z(x, y, z, t) = 2sin(x)sin(y)cos(z)
-
-The corresponding implementation will be::
-
- # user defined functions
- def func2d(x, y, t):
- vx = cos(x)
- vy = cos(y) + x*t
- return vx, vy
-
- def w_TG(res, x, y, z, t):
- res[0][...] = - cos(x) * sin(y) * sin(z)
- res[1][...] = - sin(x) * cos(y) * sin(z)
- res[2][...] = 2. * sin(x) * sin(y) * cos(z)
- return res
-
- # build a 2d domain
- box2d = Box(length=[1., 1.])
- # vector field
- v2 = Field(domain=box2d, name='v2', formula=func2d, is_vector=True,
- vectorize_formula=True)
-
- # space discretisation and data distribution description
- d2d = Discretization([65, 65])
- topo2d = box2d.create_topology(d2d)
- v2d = v2.discretize(topo2d)
- t=3.2
- v2.initialize(time=t, topo=topo2d)
- # --> call func2d for each grid point and time=0
- # to compute v2d.data[0] and v2d.data[1]
-
- # 3d
- box3d = Box()
- v3 = Field(domain=box3d, name='v3', formula=w_TG)
- # space discretisation and data distribution description
- d3d = Discretization([65, 65, 129])
- d3d_bis = Discretization([23, 65, 19], [2, 2, 2])
- topo3d = box3d.create_topology(d3d)
- topo3d_bis = box3d.create_topology(d3d_bis)
- v3d = v3.discretize(topo3d)
- v3d_bis = v3.discretize(topo3d_bis)
- t=3.2
- v3.initialize(time=t)
- # --> call w_TG for each grid point and time=t
- # to compute v3d.data[0] and v3d.data[1], v3d.data[2]
- # AND v3d_bis components.
-
-
-Notice that the local mesh resolution or the data distribution are hidden from the user-defined function.
-
-
-Add extra-parameters for field'initialization
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-In addition to the standard, required, arguments of user-defined function specified above, one can
-add some parameters in the function definitions. Consider for example the definition of velo above
-and suppose that we need to introduce a dependency with the lengthes of the domain, Lx and Ly, such that:
-
-.. math::
-
- velo_x(x,y,t) &=& L_x cos(x)\\
- velo_y(x,y,t) &=& L_y cos(y) + tx
-
-This is possible using::
-
- def func2d(x, y, t, lx, ly):
- vx = lx * cos(x)
- vy = ly * cos(y) + x*t
- return vx, vy
-
- Lx = 1.
- Ly = 3.
- extras = (Lx, Ly,) # Must be a tuple!
- vec = Field(domain=box, is_vector=True, formula=func2d,
- name='v2', vectorize_formula=True)
- vec.set_formula_parameters(extras)
- # ...
- vec.initialize() # --> use lx=1, ly=3
- vec.set_formula_parameters((3., 4.))
- vec.initialize() # --> use lx=3., ly=4.
-
-
-Other ways to initialize a field
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-* Copy an existing field::
-
- vec = Field(domain=dom, is_vector=True, name='v1')
- vec2 = Field(domain=dom, is_vector=True, name='v2')
- vec.discretize(topo)
- vec.initialize()
-
- vec2.copy(vec, topo)
- # --> discretize vec2 on topo and copy vec values for topo
-
-* Random initialization::
-
- vec = Field(domain=dom, is_vector=True, name='v1')
- vec.randomize(topo)
- # --> discretize vec on topo and init discrete field with
- # random values.
-
-
-Fields' data access
--------------------
-
-Some examples on how to get access or set values of the fields::
-
- # Get a discretisation for a given continuous field:
- velo = Field(domain=box, is_vector=True, name='v1')
- # discretize and get discrete field:
- vd = velo.discretize(some_topo)
-
- # Get all the discretisations :
- for topo in velo.discreteFields.keys():
- current_field = velo.discreteFields[topo]
-
- # Get the value of the second component of velo at point of index 8,8,8
- # in some_topo discretization
- print vd[1][8,8,8]
-
-
-Writing/reading fields to/from file
------------------------------------
-
-Fields can be written to or read from hdf files::
-
- vec = Field(domain=box, is_vector=True, name='v1')
- vec.discretize(topo)
- simu = Simulation(...)
- #...
- # Write to file
- vec.hdf_dump(topo, simu)
-
- # Read from file
- vec.hdf_load(topo)
-
-
-Other useful functions
-----------------------
-
-
-* :math:`norm(v)_d = (\sum_{i,j,...} (|v_d(i,j,...)|^2) )^{0.5}`
-
- Sum over all grid points of the values of component d of the discrete field. Computed with::
-
- v1 = Field(...)
- v1.discretize(topo)
- res = v1.norm(topo)
- # res an array of size v1.nb_components
-
-* :math:`normh(v)_d = (h_d\sum_{i,j,...} (|v_d(i,j,...)|^2) )^{0.5}`
-
- h_d being the space discretisation step in direction d::
-
- v1 = Field(...)
- v1.discretize(topo)
- res = v1.normh(topo)
- # res an array of size v1.nb_components
diff --git a/docs/sphinx/users_guide/finite_differences.rst b/docs/sphinx/users_guide/finite_differences.rst
deleted file mode 100644
index ea6ad156167681ed0736ebead3fbccd094c93702..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/finite_differences.rst
+++ /dev/null
@@ -1,229 +0,0 @@
-.. _finite_differences:
-
-.. currentmodule:: hysop.numerics.finite_differences
-
-Finite differences schemes
---------------------------
-
-To differentiate some field(s) in one direction using finite differences.
-
-If tab is a numpy array representing the discrete values of a scalar field :math:`\rho`
-on a grid, then to compute
-
-.. math::
- \begin{eqnarray}
- result &=& \frac{\partial \rho}{\partial y}
- \end{eqnarray}
-
-the basic usage of such schemes is :
-
-.. code::
-
- # Build/declare the scheme for a given space discretization
- scheme = FDC2(step, indices)
- # Apply scheme on the array
- dir = 1 # y direction
- result = scheme(tab, dir, result)
-
-with :
-
-result[iout] = diff(tab[indices], dir), diff depending on
-the chosen scheme. See examples below to understand properly indices and iout.
-
-
-Available schemes
-^^^^^^^^^^^^^^^^^
-
-* :class:`~FDC2` : first derivative, 2nd order centered scheme
-* :class:`~FD2C2`: second derivative, 2nd order centered scheme
-* :class:`~FDC4`: first derivative, 4th order centered scheme
-
-
-A few important remarks
-^^^^^^^^^^^^^^^^^^^^^^^
-
-* Mind that 'tab' arg is a numpy array not a field. Usually,
- you will have to use some_field.data[i] as tab, some_field being a :class:`~hysop.fields.discrete.DiscreteField` and i the component number.
-* step is the space discretization step size in each direction, i.e. a list or numpy array
- with d values, d being the dimension of the domain. A common case is code::
-
- step = topo.mesh.space_step
-
-* indices represent the set of points on which the scheme must be applied. This is usually
- a list of slices, for example, code::
-
- indices = topo.mesh.compute_index
-
-* result must be a predefined numpy array of the 'right' size, here the same size/shape as tab.
-
-* To optimize memory usage and computational time, it's possible to reduce the size of
- the output and/or to apply the scheme on a subset of the domain.
- scheme.indices represents the indices (in topo.mesh) of points for which the derivative will be computed.
- scheme.output_indices are positions in result where the resulting derivative will be saved.
- All available possibilities are summarized through the examples below.
-
-* The size of the ghost layer depends on the scheme but is not checked! You must ensure
- that topo.ghosts() >= scheme.ghosts_layer_size.
-
-* For some schemes a work array may be provided during call. If so, it must be a numpy array of
- the same size as result. It's shape is not really important, since during call
- work will be reshaped to be of the same shape as result. This allows us to provide 1D array
- that may be shared between different operators/methods whatever the internal required shapes are.
-
-* Notice that most of the time, finite-differences are defined as internal methods of operators and work arrays management, indices list or ghosts layers are set/checked internally.
-
-
-Default case : apply scheme on all the domain with a full-size result
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-This is the standard case where result and input tab have the same shape.
-Computation is performed on all grid points, excluding ghosts.
-
-.. code::
-
- from hysop.numerics.finite_differences import FDC2
- # --- Domain, discretization and topology ---
- box = Box(length=[1., 1.])
- d2d = Discretization([33, 33], [1, 1])
- topo = box.create_topology(d2d)
- # --- Scalar field on the domain ---
- rho = Field(box, name='rho')
- # discretize and initialize rho
- rd = rho.randomize(topo)
- # --- Grid parameters required for FD scheme ---
- # Get 'computational' points, i.e. grid points excluding ghosts
- ic = topo.mesh.compute_index
- # space step
- step = topo.mesh.space_step
- # field resolution on the grid defined by topo
- shape = topo.mesh.resolution
- # create a new array to save results
- result = npw.zeros(shape)
- # --- Create FD scheme ---
- scheme = FDC2(step, ic)
- assert (topo.ghosts() >= scheme.ghosts_layer_size).all()
- # --- Apply FD scheme ---
- # compute first derivative of scal on all points (except ghosts)
- # of the 'topo' discretization of the domain
- result = scheme(rd.data[0], 1, result)
-
-In that case:
-
-* result.shape = (34, 34)
-* scheme.indices = [slice(1,33), slice(1,33)] = topo.mesh.compute_index
-* scheme.output_indices = scheme.indices
-
-Notice that ghost points of result are not updated!
-
-Apply scheme on all the domain with a reduced result
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Suppose that you want to compute a derivative on all the domain but
-use non-standard shape array to save the result. This may be useful
-if you do not want to allocate memory for ghost points in result.
-
-.. code::
-
- # Create a 'reduced-shape' result
- # result shape == shape of topo.mesh without ghosts
- shape = np.asarray(topo.mesh.resolution).copy()
- shape -= 2 * topo.ghosts()
- shape = tuple(shape)
- result = npw.zeros(shape)
- # build and apply the scheme
- scheme = FDC2(step, ic, output_indices=True)
- result = scheme(rd.data[0], 1, result)
-
-In that case:
-
-* result.shape = (32,32)
-* scheme.indices = [slice(1,33), slice(1,33)]
-* scheme.output_indices = [slice(0,32), slice(0,32)]
-
-
-Apply scheme on a subset of the domain
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. code::
-
- # We define a subset of the domain,
- # a box of size half of the domain size.
- from hysop.domain.subsets import SubBox
- sl = topo.domain.length * 0.5
- orig = topo.domain.origin + 0.1 * topo.domain.length
- subbox = SubBox(parent=topo.domain, origin=orig, length=sl)
- # then we get indices of points inside this subdomain, for
- # the discretization 'topo'
- indices = subbox.discretize(topo)[0]
- # and build a FD scheme for this set of indices
- scheme = FDC2(step, indices)
- result = npw.zeros_like(rd.data[0])
- result = scheme(rd.data[0], 1, result)
-
-In that case derivative is computed only inside subdomain and
-only the corresponding subpart of result is updated:
-
-* result.shape = (34,34), i.e. same shape as the complete mesh of topo.
-* scheme.indices = [slice(4,21), slice(4,21)]
-* scheme.output_indices = scheme.indices
-
-
-Apply scheme on a subset of the domain with a reduced-size result
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Same as above but to limit memory print, a reduced-shape result
-is used
-
-.. code::
-
- # We define a subset of the domain,
- # a box of size half of the domain size.
- from hysop.domain.subsets import SubBox
- sl = topo.domain.length * 0.5
- orig = topo.domain.origin + 0.1 * topo.domain.length
- subbox = SubBox(parent=topo.domain, origin=orig, length=sl)
- indices = subbox.discretize(topo)[0]
- # build and apply a FD scheme, notice the output_indices arg!
- scheme = FDC2(step, indices, output_indices=True)
- # result shape is computed using discretized subdomain shape
- shape = subbox.mesh[topo].resolution
- result = npw.zeros(shape)
- result = scheme(rd.data[0], 1, result)
-
-In that case:
-
-* result.shape = (17,17)
-* scheme.indices = [slice(4,21), slice(4,21)]
-* scheme.output_indices = [slice(0,17), slice(0,17)]
-
-
-Apply scheme on a subset of the domain with a predifined-size result
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-Here you can choose explicitely where in result you want to put
-the values of the computed derivative.
-
-
-.. code::
-
- # We define a subset of the domain,
- # a box of size half of the domain size.
- from hysop.domain.subsets import SubBox
- sl = topo.domain.length * 0.5
- orig = topo.domain.origin + 0.1 * topo.domain.length
- subbox = SubBox(parent=topo.domain, origin=orig, length=sl)
- indices = subbox.discretize(topo)[0]
- shape = subbox.mesh[topo].resolution
- result = npw.zeros_like(rd.data[0])
- # output_indices are explicitely given
- iout = [slice(2, 19), slice(3, 20)]
- scheme = FDC2(step, indices, output_indices=iout)
- result = scheme(rd.data[0], 1, result)
-
-In that case:
-
-* result.shape = (17,17)
-* scheme.indices = [slice(4,21), slice(4,21)]
-* scheme.output_indices = iout = [slice(2, 19), slice(3, 20)]
-
-This option is useful when input tab is a work array, with a size different
-from the topology resolution, and the result a field defined on the whole grid.
diff --git a/docs/sphinx/users_guide/forces.rst b/docs/sphinx/users_guide/forces.rst
deleted file mode 100644
index 396cf8e3a1bb8a4160fb3c9c589e24e0640ec3ed..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/forces.rst
+++ /dev/null
@@ -1,93 +0,0 @@
-.. _forces:
-
-Drag and lift computation
-==========================
-
-.. currentmodule:: hysop.operator.drag_and_lift
-
-Two operators are dedicated to drag and lift computation: :class:`MomentumForces` and :class:`NocaForces`.
-Both formulations are detailed below.
-
-
-SHOULD BE REVIEWED
-
-Momentum formula
------------------
-
-See :cite:`Chloe-thesis` for more details.
-
-The :class:`Momentum` may be used to compute Drag and lift forces as :
-
-.. math::
-
- Forces = Coeff * \frac{d}{dt}\int_O (u_k - \tilde u) dt
-
-O is the obstacle inside the flow and :math:`\tilde u` is the velocity inside the obstacle (i.e. the penalised velocity), such that :
-
-.. math::
-
- \tilde v = \chi_O\frac{v_k}{1 + \lambda dt}
-
-where :math:`v_k` is the current velocity, dt the time step, :math:`\lambda` a penalisation coefficient, with :math:`\chi` the indicator function of the obstacle:
-
-.. math::
-
- \chi_O(x) = \left\{
- \begin{align}
- 1 & \ if x \in O\\
- 0 & \ if x \notin O
- \end{align}\right. \ \forall x \in \Omega
-
-Notice that this formula depends only on the velocity and on the obstacle but requires a penalisation.
-
-The velocity is not impacted by this operator, a temporary buffer is used for all penalisation operations.
-
-Noca's formula
-----------------
-
-See :cite:`Noca-1999` or even :cite:`Noca-thesis` for the whole details about Noca's formulation.
-
-All Noca's formula are implemented in HySoP as
-
- * :class:`NocaI` for the "impulse equation"
- * :class:`NocaII` for the "momentum equation"
- * :class:`NocaIII` for the "flux equation"
-
-
-
-Usage
------
-
-.. code::
-
- # Build and init
- force_1 = MomentumForces(velocity=velo, discretization=topo,
- obstacles=obst, penalisation_coeff=[1e8],
- io_params=IOParams('Momentum'),
- normalization=some_coeff)
- # Noca : choose formulation = 1, 2 or 3
- force_2 = MomentumForces(velocity=velo, vorticity=vorti, discretization=topo,
- nu=0.3, obstacles=obst, formulation=1,
- normalization=some_coeff)
- force_1.discretize()
- force_2.discretize()
- force_1.setup()
- force_2.setup()
-
- # Compute drag and lift ...
- force1.apply(simu)
- force2.apply(simu)
-
- print force1.forces()
- print force2.drag()
- print force1.lift()
-
-
-Notes :
-
- * A penalisation coefficient is required for MomentumForces.
- * obstacle must be a list of :class:`~hysop.domain.subsets.Subset` but is
- optional for Noca formulations.
- * For Noca, a volume of control may be provided to define the domain used to compute
- the forces. By default a box of about 90% of the size of the domain is used.
- * As usual some examples may be found in operator/tests/test_drag_and_lift.py
diff --git a/docs/sphinx/users_guide/index.rst b/docs/sphinx/users_guide/index.rst
deleted file mode 100644
index c5710cc2a0dda7f6b62582c759ae06408c6725b1..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/index.rst
+++ /dev/null
@@ -1,21 +0,0 @@
-.. _user_guide:
-
-
-############################
-HySoP User Guide
-############################
-
-
-.. toctree::
- :maxdepth: 8
-
- introduction
- space
- fields
- operators
- operators_io
- topologies
- numerics
- tools
- tasks
- biblio
diff --git a/docs/sphinx/users_guide/introduction.rst b/docs/sphinx/users_guide/introduction.rst
deleted file mode 100644
index af16a0cb11dac0c77eaa22ae69b3caf9c97229d8..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/introduction.rst
+++ /dev/null
@@ -1,19 +0,0 @@
-.. _introduction:
-
-Introduction
-============
-
-HySoP (Hybrid SimulatiOn with Particles) is a library dedicated to
-flow simulation based on particles methods,
-on hybrid (multi CPU-GPU) architecture.
-
-It is written in Python (high level functionnalities) and Fortran.
-
-This document presents the main concepts and functionnalities of the software.
-
-..
- The whole classes and functions documentation is available in auto-generated
- Doxygen pages of `Numerics <http://siconos.gforge.inria.fr/Numerics/>`_
- and `Kernel <http://siconos.gforge.inria.fr/Kernel>`_ for C++ or in
- `Front-End pages <http://siconos.gforge.inria.fr/FrontEnd/html/>`_ for
- the python interface.
diff --git a/docs/sphinx/users_guide/io_utils.rst b/docs/sphinx/users_guide/io_utils.rst
deleted file mode 100644
index bc7266803eb87d49d3ea05eff31e3944c66056ec..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/io_utils.rst
+++ /dev/null
@@ -1,149 +0,0 @@
-.. _io_utils:
-
-
-I/O management: tools and conventions in HySoP
-==============================================
-
-Different tools are available in HySoP to deal with file reading or writing, for:
-
-* scalar or vector fields, in hdf files,
-* basic text (ascii) files, for example to write the time evolution of a specific variable.
-
-
-.. _file_descriptor:
-
-Files descriptors: :class:`~hysop.tools.io_utils.IOParams`
-----------------------------------------------------------
-
-For I/O operations, especially in a parallel context, some specfici parameters must be properly defined.
-In HySoP, :class:`~hysop.tools.io_utils.IOParams` object is devoted to this task.
-
-Whenever you need to access to a file, you must use IOParams. Most of the time, you can rely on the default behavior and just specify a file name, like this::
-
- from hysop import IOParams, IO
- iop = IOParams('myfile.h5')
- # use iop as input argument for i/o operators
- # ...
- # or to explicitely set the file type:
- iop2 = IOParams('f2.h5', fileformat=IO.HDF5)
- iop3 = IOParams('f3.txt', fileformat=IO.ASCII)
-
-In any case, a default path will be set, the file created if needed, an mpi communicator associated with it and so on.
-
-
-Let us now get more into details about IOParams. Such an object defines uniquely:
-
-* **IOParams.filename**, name of the output/input file (relative or absolute)
-* **IOParams.filepath**: full path to filename. Ignored if filename is absolute.
-* **IOParams.frequency**: frequency of access to this file (e.g. if equal to N, print or read every N time steps),
- default = 1
-* **IOParams.fileformat**: the chosen format (ASCII, HDF ...), default=IO.HDF5
-* **IOParams.io_leader**, the rank (in communicator of the object using IOParams) of the mpi process that will lead the read/write process, default=0.
-
-Filename is the only required parameter, other ones have default values as shown above.
-
-If filename is 'some_relative_path/name', then filepath=$PWD/some_relative_path/name.
-
-If filename is 'just_a_name', then filepath=default_path/just_a_name
-where default_path depends on the way you run python (interactivly, ipython ...)
-and is equal to :class:`~hysop.tools.io_utils.IO.default_path()` (see table below). To check your current value, try::
-
- from hysop.tools.io_utils import IO
- print IO.default_path()
-
-
-=================== =======================
-Type of execution Default path values
-=================== =======================
-interactive $PWD/interactive/p1
-command line $PWD/pythonfile_name/pN
-=================== =======================
-
-N stands for the number of mpi processes used, and $PWD for current directory.
-For example, if you run in /tmp/hysop::
-
- mpirun -np 4 myscript.py
-
-default path will be /tmp/hysop/myscript/p4
-
-
-Anyway, since the behavior of default_path strongly depends on your python version, the way you run your scropt and others things, it may be a good idea to explicitely set default path at the beginning of your
-python main program, using::
-
- from hysop import IO
- IO.set_default_path('somewhere')
-
-Notice that even in this case, pN will be appended to 'somewhere'.
-
-Examples::
-
- from hysop import IOParams
- # minimal config
- io1 = IOParams('toto.h5')
- print io1
-
- io2 = IOParams('/tmp/myexamples/tutu.h5')
- io3 = IOParams('tutu.h5', filepath='/tmp/myexamples/', frequency=4)
-
-
-As usual, check tools/tests/test_io.py for an overview of all behaviors.
-
-
-About HDF5 files in HySoP
--------------------------
-
-
-HDF5 is, among other things, a portable file format well fitted and efficient to deal with parallel I/O for high volume of data. In HySoP, the python interface h5py is used.
-
-References :
-
-* https://www.hdfgroup.org/HDF5/
-* http://www.h5py.org
-
-
-Here is a short reminder of the basic things you need to know about hdf5 in HySoP.
-
-
-A **dataset** is a multidimensionnal array of data, in our case the values of a scalar field (or of a component of a vector field) on a grid. Each dataset may have some attributes (name ...)
-
-An **hdf file** is a container of datasets, with all the associated meta-data. It is thus important to notice that a single hdf file contains all the informations concerning the mesh on which datasets are defined, the name of the fields, \ldots and is in some way self-sufficient.
-
-
-File name convention
-^^^^^^^^^^^^^^^^^^^^
-
-In HySoP, hdf files are always named 'somename.h5'. Each file can contain several datasets but corresponds to
-one and only one instant of simulation.
-
-When fields are written at several time instants (see for example :class:`~hysop.operator.hdf_io.HDF_Writer`)
-during simulation, hdf files are named 'somename_XXXXX.h5' and a 'somename.xmf' file is created.
-
-XXXXX is the iteration number which corresponds to the time when the file has been written. The xmf file gathered all h5 files, i.e the complete collection of field values at all time steps and can be post-processed with Paraview or any similar tool.
-
-
-h5 file can be 'visualized' with h5dump or HDFView or uploaded into python (see h5py doc).
-
-Saving fields to HDF files
-==========================
-
-There are two ways to write the components of a field into an hdf file. Either with the field method :class:`~hysop.fields.continuous.Filed.hdf_dump` or
-with the operator :class:`~hysop.operator.hdf_io.HDF_Writer`.
-
-Let us start with an example::
-
- from hysop import *
- dom = Box()
- d3d = Discretization([65, 65, 65])
- f = Field(dom, name='f1', is_vector=True)
- topo = dom.create_topology(d3d)
- simu = Simulation(nb_iter=4)
- simu.initialize()
- f.hdf_dump(topo, simu)
-
-Executing this will create two files named f1.xmf and f1_00000.h5
-
-The output is done for:
-
-* a given space discretisation of the field (i.e. a :class:`~hysop.mpi.topology.Cartesian`), topo parameter in the example,
-* the current values of the field, i.e. for the current simulation time, from simu parameter in the example
-* some IOParams (see :ref:`file_descriptors`), not specified in the example, default values are used (see below)
diff --git a/docs/sphinx/users_guide/mpi_utils.rst b/docs/sphinx/users_guide/mpi_utils.rst
deleted file mode 100644
index c08429ed9aa7b1d732227d76192528daab13f9be..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/mpi_utils.rst
+++ /dev/null
@@ -1,116 +0,0 @@
-.. _mpi_utils:
-
-MPI interface and tools in HySoP
-================================
-
-HySoP is able to run on multi-core/multi-nodes architectures, with distributed memory systems and is based on Message Passing Interface (MPI) standard.
-For a standard usage of the software, you do not need a deep understanding of MPI standard, and even no understanding at all.
-
-Anyway, this part is a reminder of a few things to know for a more advanced usage of HySoP parallel capabilities.
-You can skip most of it if you are at ease with mpi.
-
-Glossary
---------
-
-HySoP is able to run on hybrid systems, multi-nodes, multi-cores, multi-gpu.
-Let us then start with some clarifications about the vocabulary used in this documentation, concerning those hybrid architectures.
-
-**cluster** : a set of connected computers, all the available ressources for your simulation.
-
-**core** : a processing unit with its own memory (cache)
-
-**socket** or processor: a single computing component with two or more cores
-
-**node** : a single computer inside a cluster.
-
-An example of a cluster is shown below.
-
-.. image:: /figures/cluster_general.*
-
-What is important to notice is that there are different memory areas (cache, RAM ...), shared or not between cores, sockets ...
-
-
-MPI implementation
-------------------
-
-HySoP is based on mpi4py which is the Python package implementing the MPI standard.
-
-References:
-
-* MPI standard : http://www.mpi-forum.org/docs/docs.html
-* About mpi4py : http://pythonhosted.org/mpi4py/
-
-Once again, a glossary is required to clearly defined standard concepts used in this documentation, concerning MPI.
-
-**MPI process**: an instance of a running program. When executing a python programme as::
-
- mpirun -np 4 python yourscript.py
-
-it means you launch 4 autonomous 'processes', each of them executing its own code from the instructions in yourscript.py. Each process executes in its own address space, its own memory, but has the ability to communicate with the others via mpi communication routines calls. Communicate means more or less to read or write some data from/to another process memory. The number of mpi processes is not supposed to correspond to the number of real physical cores available on the cluster but it's certainly not a good idea to use a number of mpi processes greater than the number of available physical cores. The mapping between processes and cores (that may be anywhere on the cluster) is done by default by MPI but may be driven by user.
-
-**Communicator**: a group (i.e. an ordered set) of MPI processes, inside a 'context' of communication.
-The context defines the properties of the communicator and how communications occur between the processes of the communicator. Mind that a message sent in one context cannot be received in another context.
-
-**Rank** : integer id of a process, relative to a given communicator. Mind that the same process may have different
-ranks in two different communicators. Ranks always start at 0.
-
-**Task** : a 'job' to be done by a set of processes. For example, consider the case where you need to solve
-an advection problem, i.e. find the values of a scalar field in a given velocity field and another problem (Navier-Stokes not to mention it) used to find the velocity field. Then, two **tasks** can be defined: one for the advection problem, attributed to one mpi process which may drive a GPU, and another task, attributed to all other processes running in parallel to solve Navier-Stokes.
-
-
-hysop and mpi
--------------
-
-When running a simulation with hysop, a default communicator is used, including all the processes attributed at runtime
-(by mpirun command), namely main_comm. The rank of each process in main_comm is main_rank::
-
- from hysop.mpi import main_comm, main_rank, main_size
- print 'I am process number ', main_rank, ' among ', main_size
-
-Running a script containing the previous lines with::
-
- mpirun -np 4 python myscript.py
-
-ended in::
-
- I am process number 1 among 4
- I am process number 2 among 4
- I am process number 3 among 4
- I am process number 0 among 4
-
-
-The mpi4py implementation is hidden into hysop.mpi package. To call any mpi4py function, try::
-
- from hysop.mpi import MPI as MPI
- # example : get main communicator size:
- MPI.COMM_WORLD.Get_size()
-
-
-mpi parameters
---------------
-
-To pass mpi-related arguments in many functions and classes in HySoP, a specific object is used, namely :class:`~hysop.tools.parameters.MPIParams`.
-
-with the following attributes:
-
-* comm : a communicator, default=main_comm
-* task_id : ... id of a task, default=DEFAULT_TASK_ID
-* rank : current process rank in comm, default=main_rank
-* on_task: boolean, true if the task_id of the current process is task_id, default=True
-
-
-Example::
-
- from hysop import MPIParams
- # default
- mp = MPIParams()
-
- # new sub-communicator:
- comm = ... # create a subcomm
- mp = MPIParams(comm)
-
-
-Most of the time, you will not have to set this parameter and can rely on default behavior.
-It's only useful when several tasks are required.
-
-Communicator splitting, task creation and so on are domain'jobs and the process is detailed in :ref:`domains`.
diff --git a/docs/sphinx/users_guide/numerics.rst b/docs/sphinx/users_guide/numerics.rst
deleted file mode 100644
index 642bb129fcf2fcd21d67424fad3c8b46c4db8f30..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/numerics.rst
+++ /dev/null
@@ -1,14 +0,0 @@
- .. _numerics:
-
-Numerical operations
-====================
-
-This module handles all the 'low-level' operations mostly applied on numpy arrays.
-
-.. toctree::
- :maxdepth: 2
-
- finite_differences
- differential_operations
- remeshing_and_interpolation
- odesolvers
diff --git a/docs/sphinx/users_guide/odesolvers.rst b/docs/sphinx/users_guide/odesolvers.rst
deleted file mode 100644
index 1221ebbb4f8e4971b904a30f641c3e6b2d9f0d49..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/odesolvers.rst
+++ /dev/null
@@ -1,51 +0,0 @@
-.. _odesolvers:
-
-ODE solvers
-===========
-
-.. currentmodule hysop.numerics
-
-Consider the following problem:
-
-.. math::
-
- \frac{d y(x,t)}{dt} &=& rhs(t, y(x,t)) \\
- y(x, t_0) &=& y_0
-
-
-with y and rhs some vector fields defined on a 1,2 or 3D domain.
-
-The following time integrators are implemented in hysop:
-
-* forward Euler, :class:`~hysop.numerics.odesolvers.Euler`,
-* forward Runge-Kutta of order 2, 3 and 4 : :class:`~hysop.numerics.odesolvers.RK2`, :class:`~hysop.numerics.odesolvers.RK3`, :class:`~hysop.numerics.odesolvers.RK4`
-
-Usage
------
-
-::
-
- # Build the system with 'some_func' as right-hand side
- ti = Euler(nb_components=2, topo=topo, f=some_func)
-
- # integrate between t and t+dt, with y representing y(x,t) on topo,
- # save y(x,t+dt) in result:
- result = ti(t, y, dt, result)
-
-
-
-To build a time integrator, it is necessary to define a python function to compute the right-hand side.
-This function must be of the following form::
-
- def some_func(t, y, work):
- work[0][...] = ...
- work[1][...] = ...
- ...
- return work
-
-
-* nb_components being the number of components of the right-hand side.
-* y and work some lists of numpy arrays of length nb_components
-* t is the time
-* y must not be modified by the function
-* work must contain the result of the computation
diff --git a/docs/sphinx/users_guide/operators.rst b/docs/sphinx/users_guide/operators.rst
deleted file mode 100644
index b225fa9cc1857a24e78541b1f27efb6a713a6989..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/operators.rst
+++ /dev/null
@@ -1,120 +0,0 @@
- .. _operators:
-
-Operators in HySoP, basic ideas
-================================
-
-This is an introduction about all kinds of operators available in HySoP with the basic ideas
-and common tools shared between all operators. You should read this whole section before any attempt
-to use a specific operator.
-
-Continuous operators
---------------------
-
-A **continuous operator** is an object that handles a set of continuous fields, depending on time and space,
-with an 'apply' function that may modify some of these variables.
-
-For example, consider the advection of a scalar :math:`\rho` at velocity v, as defined in the following system
-
-.. math::
-
- \frac{\partial{\rho(x, t}}{\partial{t}} + v(x,t).\nabla \rho(x,t) = 0
-
-To simulate this problem, an advection operator must be defined as Code::
-
- # v and rho some pre-defined continuous fields
- advec = hysop.operator.advection(v, rho, ...)
- advec.discretize()
- advec.setup()
- ...
- advec.apply()
-
-Obviously, many other parameters are required, such as which methods will be used for the space discretisation, the time integration and so on. This depends on each operator and will be detailed later.
-
-An operator is determined by
-
-* its name which sets the 'type' of problem it handles,
-* a list of variables, the continuous fields, defined on one and only one domain,
-* an mpi-task,
-* some i/o parameters.
-
-.. toctree::
- :maxdepth: 2
-
-
-Variables
-^^^^^^^^^
-
-Each operator is associated with a set of continuous fields splitted into 'input' (read only) and
-'output' (read and write) variables.
-
-.. toctree::
- :maxdepth: 2
-discrete_op
- operators_io
-
-Computational operators
------------------------
-
-Methods
-^^^^^^^
-
-Method argument must be a dictionnary with some of the following keys/values:
-
-Methods have default values, taken from hysop.default_methods
-preset dictionnaries.
-
-Keys in methods dict are given in hysop.method_keys.
-
-
-Discretisation and setup
-^^^^^^^^^^^^^^^^^^^^^^^^
-
-Each computational operator has a 'discretize' method, which:
-
-* check if each continuous field of the operator (in variables) is associated with either a :class:`~hysop.tools.parameters.Discretization` or a :class:`~hysop.mpi.topology.Cartesian`
-* discretize, if needed, each continuous field,
-
-and a 'setup' function which creates a discrete operator and ensure that the operator is ready for simulation. This function handles also internal work arrays management (see below).
-
-So, the correct and required sequence for each computational operator is::
-
- # build
- op = Operator(...)
- # discretize
- op.discretize()
- # make it ready for apply
- op.setup(rwork=..., iwork=...)
-
-.. toctree::
- :maxdepth: 2
-
- work_arrays
-
-
-Data-distribution operators
----------------------------
-
-Discrete operators
-------------------
-
-A **discrete operator** is the representation of the space discretisation of a continuous operator, i.e.
-a discrete object which applies on some discrete fields. Such an object is associated with at least one
-topology (but maybe several for multi-resolution operators).
-
-
-Review
-------
-
-.. toctree::
- :maxdepth: 2
-
- absorption_bc
- poisson
- diffusion
- forces
- penalisation
- adaptive_time_step
- advection
- stretching
- energy_enstrophy
- custom
diff --git a/docs/sphinx/users_guide/operators_io.rst b/docs/sphinx/users_guide/operators_io.rst
deleted file mode 100644
index c87976a8083baf1ef63940eda41460326eb7b05b..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/operators_io.rst
+++ /dev/null
@@ -1,8 +0,0 @@
-.. _operators_io:
-
-
-Setting I/O for operators
-=========================
-
-
-Todo : explain how to set I/O for operators using Writer class ...
diff --git a/docs/sphinx/users_guide/penalisation.rst b/docs/sphinx/users_guide/penalisation.rst
deleted file mode 100644
index eef6a2eebd8e2f633817ec39aa37d2d5cb4efc11..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/penalisation.rst
+++ /dev/null
@@ -1,58 +0,0 @@
-.. _penalisation:
-
-Penalisation
-============
-
-.. currentmodule:: hysop.operator.penalization
-
-:math:`u(X,t)` is the current velocity field, :math:`\omega(X,t)` is the current vorticity field, :math:`u_D` the velocity of the obstacle, dt the time step.
-
-Velocity penalisation
----------------------
-
-:class:`Penalization`
-
-Solves:
-
-.. math::
-
- \frac{\partial u(X, t)}{\partial t} = \chi_O(X)\lambda(u_D - u(X, t))
-
-where :math:`\lambda` a penalisation coefficient, and :math:`\chi` the indicator function of the obstacle:
-
-.. math::
-
- \chi_O(X) = \left\{
- \begin{align}
- 1 & \ if X \in O\\
- 0 & \ if X \notin O
- \end{align}\right. \ \forall X \in \Omega
-
-using implicit Euler method, we get:
-
-.. math::
-
- u_{k+1}(X) = \frac{u_k(X)}{1 + \chi_O(X)\lambda dt} + \frac{\chi_O(X)\lambda dt}{1 + \chi_O(X)\lambda dt}u_D
-
-After that, vorticity is computing as:
-
-.. math::
-
- \omega(X,t) = \nabla \times u(X,t)
-
-Vorticity penalisation
-----------------------
-
-:class:`PenalizeVorticity`
-
-Solves:
-
-.. math::
-
- \frac{\partial \omega(X, t)}{\partial t} = \nabla \times \left[\chi_O(X)\lambda(u_D - u(X, t))\right]
-
-which leads to
-
-.. math::
-
- \omega_{k+1}(X) = \omega_k(X) + \nabla \times \left[\frac{- \chi_O(X)\lambda dt}{1 + \chi_O(X)\lambda dt}(u_k(X) - u_D)\right]
diff --git a/docs/sphinx/users_guide/poisson.rst b/docs/sphinx/users_guide/poisson.rst
deleted file mode 100644
index 1e1613de8c20fcb799b05d9f01405745dc6a0505..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/poisson.rst
+++ /dev/null
@@ -1,31 +0,0 @@
-.. _poisson:
-.. currentmodule:: hysop.operator.poisson
-
-Poisson Operator
------------------
-
-.. todo::
- complete this page!!
-
-The class :class:`Poisson` is used to model and solve the following problem :
-
-.. math::
- \Delta \psi = -\omega \\
- v = \nabla \times \psi
-
-with
-
-.. math::
- \omega = \nabla \times v \\
- \nabla . v = 0
-
-
-
-Examples
-^^^^^^^^
-
-
-Code::
-
- from hysop.operator.poisson import Poisson
- op = Poisson(...)
diff --git a/docs/sphinx/users_guide/redistribute.rst b/docs/sphinx/users_guide/redistribute.rst
deleted file mode 100644
index c1c383fb68c176cbd90b0c799cf80b7c69da2de5..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/redistribute.rst
+++ /dev/null
@@ -1,101 +0,0 @@
-.. _redistribute:
-
-Data transfers between topologies
-=================================
-
-A continuous field may be associated with different topologies (in the sense of different space discretization and/or mpi processes distribution)
-depending on in which operators this field is involved. Consider for example a problem with stretching and Poisson operators, in 3 dimension, with the following sequence:
-
-* vorticity = stretching(vorticity, velocity), with a topology 'topo_s'
-* velocity = poisson(vorticity), with a topology 'topo_p'
-
-For the first operator (stretching), the best data distribution will be a 3d mpi processes grid, with a space discretization including ghost points (to fit with finite different scheme requirements), while if Poisson is solved with fft, a 1 or 2D mpi processes grid will be required, and ghost points are not necessary.
-Therefore, fields present in both operators will be associated with two different topologies. Vorticity output from stretching will be used as input in Poisson. Since data distribution and local meshes won't be the same for stretching and poisson, vorticity data need to be distributed from topo_s to topo_p. This is the role
-of :class:`~hysop.operator.redistribute.Redistribute` operator.
-
-The correct sequence to solve the problem will be:
-
-* vorticity = stretching(vorticity, velocity), with a topology 'topo_s'
-* redistribute(vorticity, topo_s, topo_p)
-* velocity = poisson(vorticity), with a topology 'topo_p'
-
-Second step means 'update values of vorticity on topo_p with its values on topo_s for all mpi processes
-involved in topo_s and topo_p'
-
-Three kind of redistribution are available in HySoP:
-
-* :class:`~hysop.operator.redistribute.RedistributeIntra` : for topologies/operator defined on
- the same mpi communicator
-* :class:`~hysop.operator.redistribute.RedistributeInter` : for topologies/operator defined on
- two different communicators (with no intersection between their sets of mpi processes)
-* :class:`~hysop.operator.redistribute.RedistributeOverlap` : for topologies/operator defined on
- on two different communicators that may have some common processes.
-
-
-
-
-In addition to the standard operator arguments, (see :ref:`operators`)
-redistribute operator needs a 'source' and a 'target'. Source and target might be
-either a :class:`~hysop.mpi.topology.Cartesian` or a :class:`~hysop.operator.computational.Computational
-
-
-
-Many different topologies can coexist for a single problem. Indeed,
-As desr
-Each operator is defined on a specific topology.
-Which means that its fields are distributed through mpi processes
-and defined on local grids.
-
-Thus, when an operator provided its output field as an input
-for the next operator, it may be required to redistribute its fields to fit
-with the topology of the next operator. Redistribute operators are made for
-that.
-
-Example::
-
- op1 = Poisson(vorticity, ...)
- op2 = Stretching(vorticity, velocity)
-
-We consider a global resolution of size :math:`N^3` and 8 mpi processes.
-
-For op1, the domain is cut into planes, and data are distributed
-through only one direction. Local grids will be of size :math:`N\times N\times\frac{N}{8}`.
-
-For op2, a 3D topology will be at work and data are distributed through
-all directions. Local grids will be of size :math:`\frac{N}{2}^3`.
-
-So, on each mpi process, vorticity data must be distributed
-from a :math:`N\times N\times\frac{N}{8}` to a :math:`\frac{N}{2}^3` grid.
-Which implies local copies and mpi communications::
-
- op1 = Poisson(vorticity, ...)
- redis = Redistribute(op1, op2, [vorticity])
- op2 = Stretching(vorticity, velocity)
-
-
-What do we need:
-
-* redistribute some fields from one topology to another.
-
-var[source] ---> var[target]
-
-Req :
- * Both topologies on the same mpi task.
- * vars must be already discretized on each topo
-
-sign::
-
- redistr(source, target, vars)
-
-
-* redistribute the fields of an operator onto the fields of another
-
-
-sign::
-
- redistr(source, target, [vars])
-
- or
-
- redistr(source, target) ==> all variables belonging to both operators
- are distributed
diff --git a/docs/sphinx/users_guide/remeshing_and_interpolation.rst b/docs/sphinx/users_guide/remeshing_and_interpolation.rst
deleted file mode 100644
index 25ac93b0097dcda781ebc3ee5fc99a797bc58ea3..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/remeshing_and_interpolation.rst
+++ /dev/null
@@ -1,59 +0,0 @@
-.. _remesh_and_interpolate:
-
-
-Interpolation and remeshing methods
-===================================
-
-Interpolation
--------------
-
-Use :class:`~hysop.numerics.interpolation.Linear to interpolate a field (scalar or vector)
-from one mesh to another.
-
-
-:class:`~hysop.numerics.interpolation.Interpolation
-
-**What**: interpolation of a field, defined as a numpy array from one "grid" to another.
-
-To interpolate a field, the following inputs are required:
-
-* a 'source' grid, provided as a :class:`~hysop.mpi.topology.Cartesian`
-* a 'target' grid, i.e. the coordinates of the points on which the field will be interpolated.
- May be defined with a :class:`~hysop.mpi.topology.Cartesian` or with the shape of a mesh.
-
-
-
-
-* requirements for position parameter
-
- * :math:`shape[d] == topo_target.mesh.resolution[d] \forall d \neq idir` where idir is the interpolation direction.
- * :math:`shape[idir] \leq topo_target.mesh.compute_resolution[d]`
-
-
-
-Devel notes
------------
-
-topo_source :
-
-* used to compute/get the 'left points' coordinates
-* ghost points layer size must fit with interpolation scheme
-* used to compute/get space step size
-* used to compute/get domain origin
-
-
-topo_target :
-
-* used to compute work arrays shape
-* must be included into topo_source domain
-* used to compute/get list of indices of interpolated points
-
-We must take into account the three cases below:
-
-* interpolate from one grid to another, grid defined as topology. Easy case, we know both topo/
-* interpolate from one grid to 'particles', when particles are initialized everywhere on a grid.
- We also know both topo so, it works.
-* interpolate from one grid to 'particles' when a threshold is used to initialize particles.
- In that case, the set of indices of particles must be provided to the interpolation caller since
- topo_target is not known (not defined indeed). This to set the variable ic in __call__.
- THIS REMAINS TO BE DONE/TEST PROPERLY
diff --git a/docs/sphinx/users_guide/space.rst b/docs/sphinx/users_guide/space.rst
deleted file mode 100644
index 6714436f44c43a7846047bbc4d78bca71c01c241..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/space.rst
+++ /dev/null
@@ -1,14 +0,0 @@
-.. _toc_dom:
-
-Domains, data distribution and space discretisation
-===================================================
-
-
-Roughly speaking, building a problem in HySoP consists in describing some fields (vectors and/or scalars), defined on physical domains, then some operators are applied to compute the behavior of those fields in time and space. This part of the manual describes fields, domains and the way data are discretized and distributed in space.
-
-.. toctree::
- :maxdepth: 2
-
- domains
- subsets
- redistribute
diff --git a/docs/sphinx/users_guide/stretching.rst b/docs/sphinx/users_guide/stretching.rst
deleted file mode 100644
index ef2793c1d44666aec04fb3a5e9185098533ddbf3..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/stretching.rst
+++ /dev/null
@@ -1,89 +0,0 @@
-.. _stretching:
-
-Stretching
-==========
-
-.. currentmodule:: hysop.operator.stretching
-
-The stretching equation, :math:`\frac{\partial\omega}{\partial t} = (\omega.\nabla)u`
-which models the lengthening of vortices in a three-dimensional flow,
-may be given under several formulations :cite:`Cottet2000vortex`:
-
-
-.. math::
-
- \frac{\partial\omega}{\partial t} &=& [\nabla u] \cdot \omega \\
- &=& [\nabla u]^T \cdot \omega \\
- &=& \nabla \cdot (\omega:u)
-
-
-
-
-Solve
-
-Two formulations are proposed :
-
-* operator :class:`Conservative`, which corresponds to the resolution of the conservative form:
-
- .. math::
-
- \frac{\partial \omega}{\partial t} = div(\omega : u)
-
-* operator :class:`GradUW`, which corresponds to the resolution of:
-
- .. math::
-
- \frac{\partial \omega}{\partial t} = [\nabla u]^T . \omega
-
-
-The default is 'conservative'.
-
-Usage
------
-
-
-TO BE REVIEWED
-.. code::
-
- m_guv = {Formulation:'GradUW', TimeIntegrator='RK3', SpaceDiscretisation='FDC4'}
- stretch_graduv = Stretching(velocity=v, vorticity=w, discretisation=some_topo, method=m_guv
- m_cons = {Formulation:'GradUW', TimeIntegrator='RK3', SpaceDiscretisation='FDC4'}
- stretch_conservative = Stretching(velocity=v, vorticity=w, discretisation=some_topo, method=m_cons)
-
-
- stretch = Conservative(velocity=v, vorticity=w, discretisation=some_topo,
- method={Formulation:'GradUW', TimeIntegrator='RK3', SpaceDiscretisation='FDC4')
-
-
-
-Linearized Stretching
----------------------
-
-In the context of a global linear stability study of the Navier-Stokes equation, one may have to formulate and solve
-a "linearized stretching".
-
-Let us decompose the vorticity and the velocity fields into the sum of the steady state solution, :math:`(.)_b`, and a perturbation,
-:math:`(.)'`:
-
-.. math::
-
- (\omega,u) = (\omega' + \omega_b, u' + u_b)
-
-then the stretching part of the governing equations writes:
-
-.. math::
-
- \frac{\partial\omega'}{\partial t} = \underbrace{(\omega'\cdot\nabla)u_b}_{A} + \underbrace{(\omega_b\cdot\nabla)u'}_{B}
-
-
-The corresponding operator is :class:`StretchingLinearized`, built with the same parameters as for the standard stretching
-plus two fields corresponding to the steady state part of the velocity and the vorticity ('BF' below stands for 'Base Flow'):
-
-
-.. code::
-
- stretch_lin = StretchingLinearized(velocity_BF=v_b, vorticity_BF=w_b,
- velocity=v, vorticity=w, ...)
-
-
-Notice that time-integrator for part A of the equation above is user-defined but set to Euler scheme for part B.
diff --git a/docs/sphinx/users_guide/subsets.rst b/docs/sphinx/users_guide/subsets.rst
deleted file mode 100644
index 36e2d7838893ecbfbd9f6740a1e378cf9f94e721..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/subsets.rst
+++ /dev/null
@@ -1,27 +0,0 @@
-.. _subsets:
-
-Subsets and obstacles
----------------------
-
-.. currentmodule:: hysop.domain
-
-The class :class:`subsets.Subset` and its heirs allow user to define some subsets
-of a predefined domain. This may be useful to:
-
-* define specific area(s) where an operator applies (penalisation for example)
-* reduce the size of i/o by writing hdf output only for a specific area.
-* define obstacles inside the flow
-
-
-Porous obstacles
-^^^^^^^^^^^^^^^^
-
-:class:`porous.BiPole`
-
-.. image:: /figures/PolesExample.png
- :width: 400pt
-
-:class:`porous.QuadriPole`
-
-.. image:: /figures/QuadriPoleExample.png
- :width: 400pt
diff --git a/docs/sphinx/users_guide/tasks.rst b/docs/sphinx/users_guide/tasks.rst
deleted file mode 100644
index a1f2afa090386916a3fdde903f71dc99ddbd587f..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/tasks.rst
+++ /dev/null
@@ -1,113 +0,0 @@
-.. _tasks:
-
-About tasks management in HySoP
-===============================
-
-First note that tasks meaning in HySoP is not for well known "task
-parallelism".
-
-Whithout tasks, all MPI processes are participating in all operations
-in the simulation. Introducing tasks enable to subdivise the set of
-processes for some operators in the graph.
-
-There are three main use case:
-- One of the operators has only a sequential implementation while
- others can benefit from MPI parallelism
-- Some operators are using only GPUs and others are using CPU cores
- and user wants to use all the resources of a compute note without
- sharing a GPU among several MPI processes.
-- On-the-fly post-processing of the data on dedicated resources
-
-
-Setup tasks in HySoP is a 3 steps process:
-
-1. specify a `proc_tasks` array to hysop.domain.domain object
- creation.
-2. create the hysop.MPI_Params using task specification
- `task_id` **and** `on_task`.
-3. create the operators with the appropriate MPI_Params. We strongly
- advise to check the produced graph on the processes output (using
- HYSOP_VERBOSE) especially for the automatically inserted InterTasksRedistributes
-
-
-Details and limitations
------------------------
-
-1. A tasks is identified by an integer (`HYSOP_DEFAULT_TASK_ID` by
- default). Each process is involved in one or several tasks. For
- instance with 4 processes and 2 tasks (1 and 2):
-
-* One task per rank (disjoint tasks)
-
-
- proc_tasks = (1,2,2,2)
- proc_tasks = (2,1,2,2)
- proc_tasks = (1,2,1,2)
-
-
-* rank0 work on both tasks. In this case all items must be
- iterable. For the moment only nested tasks with the same root
- process is possible with largest tasks first
-
-
- proc_tasks = ((2,1),(2,),(2,),(2,))
- proc_tasks = ((2,1),(2,),(2,1),(2,))
-
-
-The following are not handled yet:
-
-
- proc_tasks = ((1,2),(2,),(2,),(2,)) # largest task not in front for rank0
- proc_tasks = ((2,),(2,1),(2,),(2,)) # root process is not the same for all tasks
- proc_tasks = ((2,1),(2,),(1,),(2,)) # non-nested tasks
- proc_tasks = ((1,),(1,),(1,2),(2,)) # non-nested tasks
-
-In case of nested tasks, the `domain.current_task()` is always the
-first task in the tasks list of a process.
-
-**Note** configurations with more than 2 tasks have not been tested.
-
-
-2. In case of disjoint tasks, a single `MPI_Params` still can be used:
-
-
- box = Box(proc_tasks=proc_tasks)
- mpi_params = MPIParams(comm=box.task_comm, task_id=box.current_task(), on_task=True)
-
-
-Otherwise one `MPI_Params` per task is recommanded:
-
-
- for t in box.all_tasks:
- mpi_params[t] = MPIParams(comm=box.get_task_comm(t), task_id=t, on_task=box.is_on_task(t))
-
-
-
-3. Automatic Inter-Tasks communications is done on graph
- building. Consider the following situation with 4 operators (op1,
- op2, op3 and endA) working on 3 fields (A, B anc C) and 2 tasks
- (taskA and taskB)
-
-
- # TaskA | TaskB
- # op1 A->A |
- # op2 A->B |
- # `--B--, // Inter-task communication step
- # | op3 B,C->A
- # ,--A--' // Inter-task communication step
- # endA A->A |
-
-
-Note : without endA operator, the second communication is not
-automatically inserted because A field is an output for task A
-and cannot be invalidated by op3 on other task. Inter-task
-invalidation in graph building is not yet implemented
-
-Note this algorithm is not working with overlapping tasks
-This is due to the fact that A is both input and output in taskA.
-A workaround is to use an other field A' and endA is then a copy
-
-
-Automatic communications are inserted on task change through operator
-list. A search for appropriate field is done iterating trough previous
-and next operators.
diff --git a/docs/sphinx/users_guide/tools.rst b/docs/sphinx/users_guide/tools.rst
deleted file mode 100644
index 6e2187f802b3954602b69121af4b1726fe7e23e6..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/tools.rst
+++ /dev/null
@@ -1,8 +0,0 @@
-.. _tools::
-
-
-.. toctree::
- :maxdepth: 5
-
- io_utils
- mpi_utils
diff --git a/docs/sphinx/users_guide/topologies.rst b/docs/sphinx/users_guide/topologies.rst
deleted file mode 100644
index 730dd0304522aa1bf4a22426e6d3bbcbf286c55e..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/topologies.rst
+++ /dev/null
@@ -1,49 +0,0 @@
-.. _topologies:
-
-MPI topologies and space discretisation
-=======================================
-
-.. currentmodule:: hysop.mpi
-
-:class:`topology.Cartesian` objects are used to described both the mpi grid layout
-and the space discretisations (global and local to each mpi process).
-
-Each :class:`topology.Cartesian` has a "mesh" attributes, an object of type :class:`~hysop.domain.mesh.Mesh`, which describes the grid on which the current mpi process works.
-
-TODO :
-
-* describe the different ways to build topologies
-* describe the way we define local and global grids.
-
-
-
-Example of grids
-----------------
-
-With 2 procs::
-
- global grid (node number): 0 1 2 3 4 5 6 7 8
-
- proc 0 (global indices): X X 0 1 2 3 X X
- (local indices) : 0 1 2 3 4 5 6 7
- proc 1 : X X 4 5 6 7 X X
- 0 1 2 3 4 5 6 7
-
- with 'X' for ghost points.
-
-* Node '8' of the global grid is not represented on local mesh,
- because of periodicity. N8 == N0
-
-* on proc 1, we have:
-
- * local resolution = 8
- * global_start = 4
- * 'computation nodes' = 2, 3, 4, 5
- * 'ghost nodes' = 0, 1, 6, 7
-
-Remarks:
-
-* all 'global' values refer to the discretization parameter.
- For example 'global start' = 2 means a point of index 2 in the
- global resolution.
-* only periodic grid are considered
diff --git a/docs/sphinx/users_guide/vorticity_projection.rst b/docs/sphinx/users_guide/vorticity_projection.rst
deleted file mode 100644
index d585245fa6d73a557e247998fb0955324978dc68..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/vorticity_projection.rst
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _vorticity_projection:
-
-
-Vorticity solenoidal projection using 3D Helmholtz decomposition
-================================================================
-
-
-To be done ...
-See french doc in latex
diff --git a/docs/sphinx/users_guide/work_arrays.rst b/docs/sphinx/users_guide/work_arrays.rst
deleted file mode 100644
index 6d075eb7a432f6fa10c3801cea9dc92d5efa0533..0000000000000000000000000000000000000000
--- a/docs/sphinx/users_guide/work_arrays.rst
+++ /dev/null
@@ -1,49 +0,0 @@
-.. _work_spaces:
-
-
-Internal work arrays management
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. currentmodule:: hysop.operator.computational
-
-Internal temporary buffers may be required by computational operators (buffers for Runge-Kutta scheme for example). By default, each operator handles its own work arrays and manage their memory.
-
-But, to improve performance and limit the required memory amount, user can provide some external
-storage for those arrays, for example to share them between several operators.
-
-Each operator has a :py:meth:`~Computational.get_work_properties` function, returning information on the required memory for internal stuff.
-This function returns a dictionnary with the following keys:
-
- * 'rwork' : parameters for internal real arrays
- * 'iwork' : parameters for internal integer arrays
-
-To each key corresponds a value which may be either 'None' if no internal memory is required
-or a list of tuples, each tuple being the shape of the required array.
-
-For example:
-
- >>> op = ...
- >>> op.discretize()
- >>> work_params = op.get_work_properties()
- >>> print work_params
- {'rwork': [(12, 12), (45, 12, 33)], 'iwork': None}
-
-
-means that op requires two real arrays, of shape (12,12) and (45, 12, 33) while no
-integer arrays is needed.
-
-Note that this function must be called after discretization, else the operator may lack some
-information about local resolution and data distribution.
-
-Then, if needed, one may set internal arrays for the operator with setup optional parameters, as in the example below:
-
- >>> myrwork = []
- >> for shape in work.params['rwork']
- ... myrwork.append(npw.zeros(shape))
-
- >>> op.setup(rwork=myrwork)
- >>> # And this work may be used by another operator
- >>> op2.setup(rwork=myrwork
-
-
-Warning: obviously, data in work arrays may ne modified at any time by the operator.
diff --git a/docs/sphinx/_static/logo_hysop_couleur.png b/docs/sphinx_minimal/_static/logo_hysop_couleur.png
similarity index 100%
rename from docs/sphinx/_static/logo_hysop_couleur.png
rename to docs/sphinx_minimal/_static/logo_hysop_couleur.png
diff --git a/docs/sphinx/conf.py.in b/docs/sphinx_minimal/conf.py.in
similarity index 94%
rename from docs/sphinx/conf.py.in
rename to docs/sphinx_minimal/conf.py.in
index db54e94f7cbb5425d0a3d81b1f49b55687f3b8d1..90e00ff4375069a5e2eb2da891751e60275d2f13 100644
--- a/docs/sphinx/conf.py.in
+++ b/docs/sphinx_minimal/conf.py.in
@@ -26,7 +26,6 @@
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
import sys
-import sphinx_bootstrap_theme
extensions = [
'sphinx.ext.autodoc',
@@ -38,7 +37,7 @@ extensions = [
'sphinx.ext.ifconfig',
'sphinx.ext.viewcode',
'sphinx.ext.napoleon',
- 'sphinx.ext.intersphinx', 'sphinxcontrib.bibtex',
+ 'sphinx.ext.intersphinx',
'sphinx.ext.inheritance_diagram',
'sphinx.ext.todo',
'sphinxcontrib.doxylink']
@@ -73,7 +72,7 @@ doxylink = {
autoclass_content = 'both'
# Add any paths that contain templates here, relative to this directory.
-templates_path = ['_templates']
+templates_path = []
# The suffix of source filenames.
source_suffix = '.rst'
@@ -148,16 +147,8 @@ todo_include_todos = True
# The theme to use for HTML and HTML Help pages. See the documentation for
# a list of builtin themes.
#html_theme = 'hysop'
-#html_theme = 'bootstrap'
html_theme = 'sphinx_rtd_theme'
-# Add any paths that contain custom themes here, relative to this directory.
-html_theme_path = ['@CMAKE_CURRENT_SOURCE_DIR@/sphinx/themes']
-html_theme_path += sphinx_bootstrap_theme.get_html_theme_path()
-# Theme options are theme-specific and customize the look and feel of a theme
-# further. For a list of options available for each theme, see the
-# documentation.
-html_theme_options = {'bootswatch_theme': "sandstone", }
# The name for this set of Sphinx documents. If None, it defaults to
# "<project> v<release> documentation".
@@ -278,7 +269,9 @@ latex_documents = [
# latex_domain_indices = True
# Bilbiography (bibtex)
-bibtex_bibfiles = ['hysop.bib']
+if '@BIBTEXFILE@' != '':
+ extensions.append('sphinxcontrib.bibtex')
+ bibtex_bibfiles = ['@BIBTEXFILE@']
# -- Options for manual page output ---------------------------------------
diff --git a/docs/sphinx/index.rst b/docs/sphinx_minimal/index.rst
similarity index 98%
rename from docs/sphinx/index.rst
rename to docs/sphinx_minimal/index.rst
index 70b5687198697516fd74a5e9308ed23d49114320..f21d7ad8bfd3e4226c96368b64f56617c2f97728 100644
--- a/docs/sphinx/index.rst
+++ b/docs/sphinx_minimal/index.rst
@@ -10,6 +10,7 @@ HySoP software: hybrid simulation with particles
:maxdepth: 3
reference/index
+ license
HySoP (Hybrid SimulatiOn with Particles) is a library dedicated to flow simulation based on particular methods, on hybrid (multi CPU-GPU) architecture.
diff --git a/docs/sphinx/license-title.rst b/docs/sphinx_minimal/license.rst
similarity index 100%
rename from docs/sphinx/license-title.rst
rename to docs/sphinx_minimal/license.rst
diff --git a/docs/sphinx/reference/fortran_api.rst b/docs/sphinx_minimal/reference/fortran_api.rst
similarity index 79%
rename from docs/sphinx/reference/fortran_api.rst
rename to docs/sphinx_minimal/reference/fortran_api.rst
index 778a15d07dd41dccdfce97b24224cf014e0bb244..c3800572e55e8784a68cbba45e48df7e4ec9dd48 100644
--- a/docs/sphinx/reference/fortran_api.rst
+++ b/docs/sphinx_minimal/reference/fortran_api.rst
@@ -8,4 +8,4 @@ to doxygen.
This includes fftw and scales' interfaces.
-See `Fortran API <../../../doxygen/html/index.html>`_.
+See `Fortran API <../doxygen/html/index.html>`_.
diff --git a/docs/sphinx/reference/index.rst b/docs/sphinx_minimal/reference/index.rst
similarity index 100%
rename from docs/sphinx/reference/index.rst
rename to docs/sphinx_minimal/reference/index.rst
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-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
- \usepackage{cancel}
-\usepackage{xcolor}
-\usepackage[left=3cm,right=3cm,top=3cm,bottom=3cm]{geometry}
-\usepackage{algorithm}
-\usepackage{algorithmic}
-
-%%%%%%%%%%%%%RACCOURCIS%%%%%%%%%%%%%%%%%%%%%%%%
-
-\newcommand{\vect}[1] {\underline{#1} \,}
-\newcommand{\nn}{\nonumber}
-\newcommand{\vnabla} {{\bf \nabla}}
-\newcommand{\vDelta} {{\bf \Delta}}
-\newcommand{\vx} {{\bf x}}
-\newcommand{\vdl} {{\bf dl}}
-\newcommand{\ds} {\displaystyle}
-\newcommand{\dsp}{\displaystyle}
-\newcommand{\be}{\begin{equation}}
-\newcommand{\ee}{\end{equation}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\bx{\mathbf{x}}
-% \input{article_rev.tex}
-
-\def\S{{\Sigma}}
-\def\by{\mathbf{y}}
-\def\bn{\mathbf{n}}
-\def\bz{\mathbf{z}}
-\def\bc{\mathbf{c}}
-\def\bs{\mathbf{s}}
-
-\def\bff{\mathbf{f}}
-\def\bV{\mathbf{V}}
-\def\bX{\mathbf{X}}
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-
-\def\bpsi{\mbox{\boldmath$\psi$}}
-\def\bxi{\mbox{\boldmath$\xi$}}
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-
-\def\bmu{\mbox{\boldmath$\mu$}}
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-\def\bpsi{\mbox{\boldmath$\psi$}}
-\def\bgamma{\mbox{\boldmath$\gamma$}}
-\def\bv{\mathbf{v}}
-\def\bu{\mathbf{u}}
-\def\obu{\overline{\mathbf{u}}}
-\def\obw{\overline{\bw}}
-
-
-\def\bnabla{\mbox{\boldmath$\nabla$}}
-\def\bDelta{\mbox{\boldmath$\Delta$}}
-\def\w{\mbox{$\omega$}}
-\def\btimes{\mbox{\boldmath$\times$}}
-\def\rot{\mbox{\boldmath$\nabla\times$}}
-\def\div{\bnabla\cdot}
-\def\cal#1{\mathcal{#1}}
-\def\Rr{\hbox{\rm I\hskip -1.5pt R}}
-\def\dpart#1#2{\frac{\partial#1}{\partial#2}}
-\def\ddpart#1#2{\frac{\partial^2#1}{\partial#2}}
-\def\ddpartden#1#2#3{\frac{\partial^2#1}{\partial#2\partial#3}}
-\def\ds{\displaystyle}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={Relation Pressure-Velocity},
- pdfauthor={Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={pressure, velocity, relation, Poisson equation},
- pdfsubject={Relation Pressure-Velocity}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Relation liant la pression et le champ de vitesse dans les équations de Navier Stokes incompressibles}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-En prenant la divergence des équations de Navier-Stokes incompressibles :
-
-\be\label{NS}
- \dpart\bu t + \bu \cdot \bnabla \bu = - \bnabla p + \dfrac{1}{\text{Re}} \bDelta \bu,
-\ee
-
-et en utilisant la condition d'incompressibilité :
-\be\label{divU}
-\text{div} \ \bu = \div \bu = 0,
-\ee
-
-on obtient l'équation de Poisson suivante permettant d'exprimer la pression en fonction du champ de vitesse :
-
-\be
-\boxed{\bDelta p = \sum \limits_{\substack{i=1 \\j=1 \\ i \neq j}}^{n}{\dpart {u_i}{x_i} \ \dpart {u_j}{x_j}} -
-\sum \limits_{\substack{i=1 \\j=1 \\ i \neq j}}^{n}{\dpart {u_i}{x_j} \ \dpart {u_j}{x_i}}}
-\ee
-%
-En 2D cette relation est donc donnée par :
-\be
-\bDelta p = 2 \left( \dpart {u_1}{x_1} \ \dpart {u_2}{x_2} - \dpart {u_1}{x_2} \ \dpart {u_2}{x_1} \right)
-\ee
-%
-Et en 3D par :
-\be\label{3D}
-\bDelta p = 2 \left( \left(\dpart {u_1}{x_1} \ \dpart {u_2}{x_2} + \dpart {u_1}{x_1} \ \dpart {u_3}{x_3} + \dpart {u_2}{x_2} \ \dpart {u_3}{x_3} \right)
-- \left(\dpart {u_1}{x_2} \ \dpart {u_2}{x_1} + \dpart {u_1}{x_3} \ \dpart {u_3}{x_1} + \dpart {u_2}{x_3} \ \dpart {u_3}{x_2} \right) \right)
-\ee
-%
-\underline{Démonstration pour $n=3$} :\\
-
-Calculons la divergence de chacun des termes de l'équation \ref{NS}.
-
-%%%%%%%%% dérivée en temps %%%%%%%%%%%%%%
-\begin{itemize}
- \item div ($\partial_t \bu$)
-\end{itemize}
-D'après la condition d'incompressibilité \ref{divU}, on a :
-\be
-\text{div} \left(\dpart\bu t \right) = \dfrac{\partial}{\partial t}(\text{div} \ \bu) = 0
-\ee
-
-%%%%%%%%% terme d'advection %%%%%%%%%%%%%%
-\begin{itemize}
- \item div ($\bu \cdot \bnabla \bu$)
-\end{itemize}
-
-\begin{align*}
-\text{div} \ (\bu \cdot \bnabla \bu) & =
-\left( \begin{array}{c}
-\partial x \\
-\partial y \\
-\partial z
-\end{array} \right)
-\cdot
-\left( \begin{array}{c}
-\vspace{0.3cm}
-u_1 \dpart{u_1}{x_1} + u_2 \dpart{u_1}{x_2} + u_3 \dpart{u_1}{x_3} \\
-\vspace{0.3cm}
-u_1 \dpart{u_2}{x_1} + u_2 \dpart{u_2}{x_2} + u_3 \dpart{u_2}{x_3} \\
-u_1 \dpart{u_3}{x_1} + u_2 \dpart{u_3}{x_2} + u_3 \dpart{u_3}{x_3}
-\end{array} \right) \\
-& = \left( \textcolor{blue}{\dpart {u_1}{x_1} \ \dpart {u_1}{x_1}} + \textcolor{red}{u_1 \ddpart {u_1}{x_1^2}} \right) +
-\left( \dpart {u_2}{x_1} \ \dpart {u_1}{x_2} + \textcolor{brown}{u_2 \dfrac {\partial^2 u_1}{\partial x_1 \partial x_2}} \right) +
-\left( \dpart {u_3}{x_1} \ \dpart {u_1}{x_3} + \textcolor{violet}{u_3 \dfrac {\partial^2 u_1}{\partial x_1 \partial x_3}} \right) \\
-& + \left( \dpart {u_1}{x_2} \ \dpart {u_2}{x_1} + \textcolor{red}{u_1 \dfrac {\partial^2 u_2}{\partial x_2 \partial x_1}} \right) +
-\left( \textcolor{blue}{\dpart {u_2}{x_2} \ \dpart {u_2}{x_2}} + \textcolor{brown}{u_2 \ddpart {u_2}{x_2^2}} \right) +
-\left( \dpart {u_3}{x_2} \ \dpart {u_2}{x_3} + \textcolor{violet}{u_3 \dfrac {\partial^2 u_2}{\partial x_2 \partial x_3}} \right) \\
-& + \left( \dpart {u_1}{x_3} \ \dpart {u_3}{x_1} + \textcolor{red}{u_1 \dfrac {\partial^2 u_3}{\partial x_3 \partial x_1}} \right) +
-\left( \dpart {u_2}{x_3} \ \dpart {u_3}{x_2} + \textcolor{brown}{u_2 \dfrac {\partial^2 u_3}{\partial x_3 \partial x_2}} \right) +
-\left( \textcolor{blue}{\dpart {u_3}{x_3} \ \dpart {u_3}{x_3}} + \textcolor{violet}{u_3 \ddpart {u_3}{x_3^2}} \right) \\
-& = \textcolor{blue}{\alpha} + \beta + \textcolor{red}{\gamma} + \textcolor{brown}{\delta} + \textcolor{violet}{\epsilon}
-\end{align*}
-%
-Avec :
-%
-\begin{align*}
-\textcolor{blue}{\alpha} & = \dpart {u_1}{x_1} \ \dpart {u_1}{x_1} + \dpart {u_2}{x_2} \ \dpart {u_2}{x_2} + \dpart {u_3}{x_3} \ \dpart {u_3}{x_3} \\
-\beta & = 2 \dpart {u_2}{x_1} \ \dpart {u_1}{x_2} + 2 \dpart {u_1}{x_3} \ \dpart {u_3}{x_1} + 2 \dpart {u_2}{x_3} \ \dpart {u_3}{x_2}\\
-\textcolor{red}{\gamma} & = u_1 \ddpart {u_1}{x_1^2} + u_1 \dfrac {\partial^2 u_2}{\partial x_2 \partial x_1} + u_1 \dfrac {\partial^2 u_3}{\partial x_3 \partial x_1} \\
-\textcolor{brown}{\delta} & = u_2 \ddpart {u_2}{x_2^2} + u_2 \dfrac {\partial^2 u_1}{\partial x_1 \partial x_2} + u_2 \dfrac {\partial^2 u_3}{\partial x_3 \partial x_2}\\
-\textcolor{violet}{\epsilon} & = u_3 \ddpart {u_3}{x_3^2} + u_3 \dfrac {\partial^2 u_1}{\partial x_1 \partial x_3} + u_3 \dfrac {\partial^2 u_2}{\partial x_2 \partial x_3}
-\end{align*}
-%
-Or :
-%
-\begin{align*}
-\alpha & =
-\dpart {u_1}{x_1} \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) - \dpart {u_1}{x_1} \ \dpart {u_2}{x_2} - \dpart {u_1}{x_1} \ \dpart {u_3}{x_3} \\
-& + \dpart {u_2}{x_2} \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) - \dpart {u_2}{x_2} \ \dpart {u_1}{x_1} - \dpart {u_2}{x_2} \ \dpart {u_3}{x_3} \\
-& + \dpart {u_3}{x_3} \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) - \dpart {u_3}{x_3} \ \dpart {u_1}{x_1} - \dpart {u_3}{x_3} \ \dpart {u_2}{x_2} \\
-& = -2 \dpart {u_1}{x_1} \ \dpart {u_2}{x_2} - 2 \dpart {u_1}{x_1} \ \dpart {u_3}{x_3} - 2 \dpart {u_2}{x_2} \ \dpart {u_3}{x_3} \\
-u_1 \ddpart {u_1}{x_1^2} & = u_1 \left( \nabla x \cdot \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) \right)- u_1 \dfrac {\partial^2 u_2}{\partial x_1 \partial x_2} - u_1 \dfrac {\partial^2 u_3}{\partial x_1 \partial x_3} \\
-& = - u_1 \dfrac {\partial^2 u_2}{\partial x_1 \partial x_2} - u_1 \dfrac {\partial^2 u_3}{\partial x_1 \partial x_3} \\
-u_2 \ddpart {u_2}{x_2^2} & = u_2 \left( \nabla y \cdot \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) \right)- u_2 \dfrac {\partial^2 u_1}{\partial x_2 \partial x_1} - u_2 \dfrac {\partial^2 u_3}{\partial x_2 \partial x_3} \\
-& = - u_2 \dfrac {\partial^2 u_1}{\partial x_2 \partial x_1} - u_2 \dfrac {\partial^2 u_3}{\partial x_2 \partial x_3} \\
-u_3 \ddpart {u_3}{x_3^2} & = u_3 \left( \nabla z \cdot \left( \cancel{\dpart {u_1}{x_1} + \dpart {u_2}{x_2} + \dpart {u_3}{x_3}} \right) \right)- u_3 \dfrac {\partial^2 u_1}{\partial x_3 \partial x_1} - u_3 \dfrac {\partial^2 u_2}{\partial x_3 \partial x_2} \\
-& = u_3 \dfrac {\partial^2 u_1}{\partial x_3 \partial x_1} - u_3 \dfrac {\partial^2 u_2}{\partial x_3 \partial x_2}
-\end{align*}
-
-Les matrices jacobiennes et hessiennes de $\bu$ sont des matrices symétriques, donc $\gamma = \delta = \epsilon = 0$.\\
-%
-Ainsi :
-%
-\begin{align*}
-\text{div} \ (\bu \cdot \bnabla \bu) &= \alpha + \beta\\
-& = \left (-2 \dpart {u_1}{x_1} \ \dpart {u_2}{x_2} - 2 \dpart {u_1}{x_1} \ \dpart {u_3}{x_3} - 2 \dpart {u_2}{x_2} \ \dpart {u_3}{x_3} \right) + \left( 2 \dpart {u_2}{x_1} \ \dpart {u_1}{x_2} + 2 \dpart {u_1}{x_3} \ \dpart {u_3}{x_1} + 2 \dpart {u_2}{x_3} \ \dpart {u_3}{x_2} \right)
-\end{align*}
-
-%%%%%%%%% terme de pression %%%%%%%%%%%%%%
-\begin{itemize}
- \item div (-$\bnabla p$) = -$\bDelta p$
-\end{itemize}
-
-%%%%%%%%% terme de diffusion %%%%%%%%%%%%%%
-\begin{itemize}
- \item div $ \left(\dfrac{1}{\text{Re}} \bDelta \bu \right) = \dfrac{1}{\text{Re}} \bDelta ( \text{div} \ \bu) = 0$
-\end{itemize}
-%
-Par conséquent :
-%
-\begin{align*}
-& \text{div} \ \left(\dpart\bu t + \bu \cdot \bnabla \bu = - \bnabla p + \dfrac{1}{\text{Re}} \bDelta \bu \right) \\
-& \Longleftrightarrow \text{div} \ (\bu \cdot \bnabla \bu) = \text{div} \ (- \bnabla p)\\
-& \Longleftrightarrow \bDelta p = \left (2 \dpart {u_1}{x_1} \ \dpart {u_2}{x_2} + 2 \dpart {u_1}{x_1} \ \dpart {u_3}{x_3} + 2 \dpart {u_2}{x_2} \ \dpart {u_3}{x_3} \right) - \left( 2 \dpart {u_2}{x_1} \ \dpart {u_1}{x_2} + 2 \dpart {u_1}{x_3} \ \dpart {u_3}{x_1} + 2 \dpart {u_2}{x_3} \ \dpart {u_3}{x_2} \right) = \ref{3D}
-\end{align*}
-\end{document}
diff --git a/docs/tex_sources/Rayleigh_Taylor_instability.pdf b/docs/tex_sources/Rayleigh_Taylor_instability.pdf
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--- a/docs/tex_sources/Rayleigh_Taylor_instability.tex
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@@ -1,182 +0,0 @@
-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\usepackage{graphics,subfigure}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
- \usepackage{cancel}
-\usepackage{xcolor}
-\usepackage[left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={RTIRMI},
- pdfauthor={Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={Rayleigh-Taylor and Richtmyer-Meshkov instabilities, baroclinic term, diphasic flows},
- pdfsubject={multiphasic instabilities}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Simulation de l'instabilité de Rayleigh-Taylor pour un écoulement incompressible}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-\section{Modèle, terme baroclinique et description physique}
-
-Dans un fluide présentant des densités différentes, un terme source apparaît dans l'équation de transport de la vorticité lorsque les surfaces de densité constante (surfaces isopycnes) et les surfaces de pression constante (surfaces isobares) ne sont pas alignées.
-Cette équation de transport de la vorticité (VTE) est obtenue en prenant le rotationnel de l'équation de Navier-Stokes en formulation vitesse-pression:
-\begin{align}\label{NS}
-{\frac {\partial {\bf{u}}}{\partial t}}+({\bf{u}}\cdot {\nabla }){ {\bf{u}} } & = \nu \Delta {\bf{u}} - \dfrac{\nabla p}{\rho} + {\bf{g}} \\
-\nabla \cdot {\bf{u}} & = 0,
-\end{align}
-
-où $\bf{u}$ est la vitesse de l'écoulement, ${\omega }$ est la vorticité, $p$ est la pression, $\rho$ est la densité et ${\bf{g}}$ la force de gravité.
-
-La VTE est alors donnée par :
-\begin{align}
-{\frac {\partial {\omega }}{\partial t}}+({\bf{u}}\cdot {\nabla }){ \omega } - (\omega \cdot {\nabla }){{\bf{u}}} &= \nu \Delta \omega - \nabla p \times \nabla \left( \dfrac{1}{\rho}\right)\\
-{\frac {\partial {\omega }}{\partial t}}+(\bf{u}\cdot {\nabla }){ \omega } - (\omega \cdot {\nabla }){ \bf{u}} &= \nu \Delta \omega + \underbrace {{\frac {1}{\rho ^{2}}}{\nabla }\rho \times {\nabla }p}_{{{\text{contribution baroclinique}}}}\label{VTEbaro}
-\end{align}
-%
-Le contribution baroclinique est donc donnée par le vecteur :
-\begin{equation}
- {\frac {1}{\rho ^{2}}}{\nabla }\rho \times {\nabla }p = \dfrac{\nabla \rho}{\rho} \times \dfrac{\nabla p}{\rho}
-\end{equation}
-%
-Afin d'éviter le calcul explicite de la pression (qui n'est pas directement nécessaire dans la formulation en vorticité de l'équation de Navier-Stokes), on peut exprimer le gradient de pression de la façon suivante, à l'aide de l'équation \ref{NS} :
-\begin{equation}
--\dfrac{\nabla p}{\rho} = {\frac {\partial {\bf{u}}}{\partial t}}+({\bf{u}}\cdot {\nabla }){ {\bf{u}} } - \nu \Delta {\bf{u}} - {\bf{g}},
-\end{equation}
-ainsi la VTE est donc alternativement donnée par :
-\begin{align}
-{\frac {\partial {\omega }}{\partial t}}+({\bf{u}}\cdot {\nabla }){ \omega } - (\omega \cdot {\nabla }){{\bf{u}}} &= \nu \Delta \omega - \dfrac{\nabla \rho}{\rho} \times \left(-\dfrac{\nabla p}{\rho} \right) \\ \label{baroNoPrs}
-{\frac {\partial {\omega }}{\partial t}}+({\bf{u}}\cdot {\nabla }){ \omega } - (\omega \cdot {\nabla }){{\bf{u}}} &= \nu \Delta \omega ~ \underbrace {- \dfrac{\nabla \rho}{\rho} \times \left({\frac {\partial {\bf{u}}}{\partial t}}+({\bf{u}}\cdot {\nabla }){ {\bf{u}} } - \nu \Delta {\bf{u}} - {\bf{g}} \right)}_{{{\text{contribution baroclinique}}}}
-\end{align}
-%
-Le vecteur baroclinique intervient aussi bien dans les écoulements compressibles que dans les écoulements incompressibles (mais hétérogènes). Les modes instables de Rayleigh–Taylor peuvent être analysés à partir de ce vecteur baroclinique. Il joue également un rôle prépondérant dans la création de vorticité par le passage de chocs à travers un milieu hétérogène, comme dans l'instabilité de Richtmeyer–Meshkov.
-
-Lorsque l'interface entre deux surfaces de densité différente n'est pas horizontale et que le système est proche de l'équilibre hydrostatique, le gradient de pression est vertical mais le gradient de densité ne l'est pas. Par conséquent, le vecteur baroclinique est non nul ce qui provoque une création de vorticité entraînant un soulèvement de l'interface.
-
-L’instabilité de Rayleigh–Taylor, est une instabilité de l’interface séparant deux fluides de densités différentes, qui résulte de la poussée du fluide le plus lourd sur le fluide le plus léger (l'accélération dans le cas d'un système dynamique ou la gravité pour un système initialement statique est dirigée vers la phase légère). Considérons deux couches de fluides non miscibles superposées dans deux plans parallèles, la plus lourde surplombant la plus légère et toutes deux soumises à la pesanteur terrestre. L’équilibre est instable à la moindre perturbation : toute perturbation va s'amplifier et libérer de l’énergie potentielle, le fluide le plus lourd gagnant progressivement la moitié inférieure sous l'effet du champ de gravitation, et le fluide léger passe au-dessus.
-
-En effet, lorsqu'on échange un volume de fluide léger, situé près de l'interface entre les deux fluides, avec un même volume de fluide lourd de l'autre côté de cette interface, l'énergie potentielle de gravitation du système diminue tout comme son énergie totale. C'est la tendance naturelle de ce système à minimiser son énergie potentielle de gravitation qui crée l'instabilité. Cette perte d'énergie est compensée par l'augmentation d'une autre forme d'énergie : l'énergie cinétique. En résumé, le fluide lourd va passer sous le léger (et le léger sur le lourd, par symétrie), et ce mouvement s'accompagne forcément d'une création d'énergie cinétique. Au contraire, dans le cas des vagues, si le fluide léger supérieur s’engouffre dans le fluide lourd inférieur, la poussée d’Archimède tend à ramener le fluide léger à sa position initiale : il y a donc au contraire stabilité et propagation d’ondes à l’interface.
-
-\section{Simulations}
-
-Les paramètres donnés dans cette section sont ceux permettant d'obtenir les résultats présentés dans la figure \ref{RTI}.
-\begin{figure}[ht]
-\centering
-\subfigure[$\rho_0 = 1, \rho_1 = 3$]
- {\includegraphics[width=0.45\linewidth]{images/diphasic/delta_rho=3.pdf}}
-\subfigure[$\rho_0 = 1, \rho_1 = 10$]
- {\includegraphics[width=0.45\linewidth]{images/diphasic/delta_rho=10.pdf}}
-\caption{Instabilité de Rayleigh-Taylor (résultats 2D de Georges-Henri).}
-\label{RTI}
-\end{figure}
-
-\begin{itemize}
- \item $\Omega = [0, \ 3] \times [0, \ 3]$ (GH), \\
-$\Omega = [-1.5, \ 1.5] \times [-1.5, \ 1.5] \times [-0.0625, \ 0.0625]$ (Parmes)
- \item res = $288 \times 288 \times 12$ (Parmes)
- \item $\Delta t = 0.005$
- \item $\nu = 10^{-4}$ (Parmes)
- \item ${\bf{u}}_{\text{init}} = (u_x, u_y, u_z)$ avec :
-\begin{align}\label{ux}
-u_x & = -\dfrac{2 y A}{\sigma \textcolor{red}{\lambda}} \ \text{cos}(\lambda) \ \text{exp}\left(-\left( \dfrac{y-y_c}{\sigma} \right)^2\right)\\
-u_y & = A \ \text{sin}(\lambda) \ \text{exp}\left(-\left( \dfrac{y-y_c}{\sigma} \right)^2\right)\\
-u_z & =0
-\end{align}
- \item $A = 0.02$
- \item $\lambda = 3 .\ 2\pi \ x / L_x$
- \item $\sigma = 0.4$
-\end{itemize}
-
-\textcolor{red}{attention} : Le paramètre $\lambda$ en rouge dans l'équation \ref{ux} n'est pas présent dans l'expression de ${\bf{u}}_{\text{init}}$ permettant d'obtenir les résultats de la figure \ref{RTI}, mais sa présence est nécessaire pour satisfaire la condition d'incompressibilité. D'autre part, ce champ de vitesse initial ${\bf{u}}_{\text{init}}$ n'est à divergence nulle que si $y_c=0$, c'est à dire si la coordonnées en $y$ du milieu du domaine est égale à 0, ce qui n'est pas le cas dans le benchmark considéré ici.
-
-\begin{itemize}
- \item $\text{log}(\rho_0) = \text{log}(1)$ si $y>y_c$\\
- $\text{log}(\rho_1) = \text{log}(3)$ ou $\text{log}(10)$ si $y \leq y_c$\\
-$\rho_0$ : densité de la phase inférieure (la plus légère) \\
-$\rho_1$ : densité de la phase supérieure (la plus dense).\\
-
-Remarque : le fait de transporter log$(\rho)$ au lieu de $\rho$ permet d'avoir une meilleure régularité au niveau de l'interface raide (qui mathématiquement n'est pas définie). En procédant de la sorte, et de part la relation $\nabla(\text{log}(\rho)) = \nabla \rho / \rho,$ la contribution baroclinique donnée dans l'équation \ref{baroNoPrs} s'exprime désormais :
-\begin{equation}
- - \nabla(\text{log}(\rho)) \times \left({\frac {\partial {\bf{u}}}{\partial t}}+({\bf{u}}\cdot {\nabla }){ {\bf{u}} } - \nu \Delta {\bf{u}} - {\bf{g}} \right).
-\end{equation}
-
-
-Une autre possibilité consiste à lisser la valeur du champ de densité au voisinage de l'interface sur une épaisseur $2\delta$ en utilisant par exemple la fonction d'erreur (aussi appelée fonction d'erreur de Gauss) $Erf$ (figure \ref{erf}) :
-\begin{equation}
-\operatorname{erf}(x)={\dfrac 2{{\sqrt \pi }}}\int _{0}^{x}e^{{-t^{2}}}\,{\mathrm d}t.
-\end{equation}
-où le champ de densité sera alors défini par :
-\begin{equation}
-\rho=\left\lbrace
-\begin{array}{lcl}
-\rho_0 ~ ~ \text{si} ~ ~ y>y_c + \delta\\
-\rho_1 ~ ~ \text{si} ~ ~ y < y_c - \delta\\
-\dfrac{\rho_1 + \rho_0}{2} \left ( 1+ \dfrac{\rho_1 - \rho_0}{2} \ \operatorname{erf} \left( \dfrac{y-y_c}{\delta} \right)\right) ~ ~ \text{si} ~ ~ \lvert y - y_c \lvert \leq \delta
-\end{array}\right.
-\end{equation}
-\end{itemize}
-
-\begin{figure}[ht]
-\centering
- \includegraphics[width=0.7\linewidth]{images/diphasic/error_function.pdf}
-\caption{Fonction d'erreur $Erf$.}
-\label{erf}
-\end{figure}
-
-\begin{figure}[ht]
-\centering
-\subfigure[transport de log$(\rho), \Lambda_2$ corrigé, $T = 19$]
- {\includegraphics[width=0.7\linewidth]{images/diphasic/RTI_rescaled_L2.png}}
-\subfigure[transport de log$(\rho), \Lambda_4$ corrigé, $T = 18$]
- {\includegraphics[width=0.7\linewidth]{images/diphasic/RTI_rescaled_L4.png}}
-\subfigure[lissage + transport de log$(\rho), \Lambda_{4,2}$, $T = 34$]
- {\includegraphics[width=0.7\linewidth]{images/diphasic/RTI_rescaled_log_liss_M6.png}}
-\subfigure[transport de log$(\rho), \Lambda_{4,2}$, $T = 2.7$]
- {\includegraphics[width=0.7\linewidth]{images/diphasic/RTI_rescaled_M6.png}}
-\caption{Resultats (PARMES)}
-\label{Res_Parmes}
-\end{figure}
-
-\begin{figure}[ht]
-\centering
-\includegraphics[width=1.0\linewidth]{images/diphasic/energie_cinetique.png}
-\label{RTI}
-\end{figure}
-
-\begin{figure}[ht]
-\centering
-\subfigure[transport de log$(\rho), \Lambda_{4,2}$]
- {\includegraphics[width=0.8\linewidth]{images/diphasic/energie_cinetique2.png}}
-\subfigure[référence Springel. $\rho_0=1$, $\rho_1=2$, \\
-$u_y{\text{init}}=0.025 \ (1 - \text{cos}(4.0 \pi x)) (1.0 - \text{cos}(4.0 \pi y / 3.0))$]
- {\includegraphics[width=0.6\linewidth]{images/diphasic/ref_energie.png}}
- {\includegraphics[width=0.95\linewidth]{images/diphasic/ref_density.png}}
-\label{RTI}
-\end{figure}
-
-\end{document}
diff --git a/docs/tex_sources/Stretching.pdf b/docs/tex_sources/Stretching.pdf
deleted file mode 100644
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deleted file mode 100644
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--- a/docs/tex_sources/Stretching.tex
+++ /dev/null
@@ -1,272 +0,0 @@
-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={Stretching-Stability},
- pdfauthor={Dalissier Eric, Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={Stretching, Stability},
- pdfsubject={Vorticity}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Condition de Stabilit\'e sur le Stretching}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-L'\'equation de Navier-Stokes peut \^etre r\'esolue en utilisant le splitting d'op\'e\-rateur, qui consiste \`a traiter ind\'ependamment les op\'erateurs du probl\`eme. Dans ce document, nous nous int\'eressons au terme de stretching qui repr\'esente les d\'eplac\'epements transversaux de l'\'ecoulement. L'\'equation de Stretching peut se pr\'esenter sous plusieurs formes~:
-\begin{align}
-\label{eq:gradstretch}
- \frac{\partial \omega}{\partial t} = [\nabla u]. \omega \\
-\label{eq:transpgradstretch}
- \frac{\partial \omega}{\partial t} = [\nabla u]^T. \omega\\
-\label{eq:divstretch}
- \frac{\partial \omega}{\partial t} = div( \omega u )
-\end{align}
-Une autre condition qui doit \^etre v\'erifi\'ee est la conservation de la masse, exprim\'ee par $div (u) =0$.
-
-La définition de la vorticit\'e est~:
-\begin{equation}
- \omega = \overrightarrow{rot}(u)
-\end{equation}
-
-
-\section{Stabilit\'e}
-L'\'equation \eqref{eq:gradstretch} conserve la masse. Cette expression peut devenir instable num\'e\-riquement, tout d\'ependra du sch\'ema d'int\'egration en temps ainsi que de la forme de la matrice $\nabla u$.
-On peut se ramener \`a l'\'etude d'un syst\`eme lin\'eaire de la forme~:
-\begin{equation}
-\label{eq:problemeglobal}
- \omega'=A \omega
-\end{equation}
-Dans le cas o\`u $A$ est diagonalisable, on peut se ramener \`a l'\'etude des valeurs propres de A, car l'on aura~:
-\begin{equation}
-\label{eq:eigentime}
-\omega' = \lambda_i \omega \hspace{0.2\linewidth} \text{ o\`u } i\in \{0,1,2\}
-\end{equation}
-
-
-On va s'int\'eresser \`a l'\'etude de la condition \`a mettre sur $\lambda$. On veut pour que le syst\`eme converge que Re$(\lambda)<1$, car c'est elle qui va entra\^iner la convergence dans la r\'esolution de l'EDO. (il faut que la puissance de l'exponentielle soit plus petite que 1 pour converger) On ne s'int\'eresse qu'\`a la partie r\'eelle car la partie complexe est une fonction oscillante entre -1 et 1, et donc bornable.
-
-On veut que l'\'equation \eqref{eq:eigentime} converge en temps, c'est-\`a-dire pour le cas de sch\'ema \`a 2 pas de temps~:
-\begin{equation}
- \omega^{n+1} = B \omega^n
-\end{equation}
-Que $B$ doit \^etre born\'ee en valeur absolue par 1 de tel sorte que~:
-\begin{equation}
-|\frac{\omega^{n+1}}{\omega^n}|<1
-\end{equation}
-
-Les conditions pour les diff\'erents sch\'emas sont les suivantes~:
-\begin{itemize}
- \item Le sch\'ema d'Euler (ordre 1)\\
-L'\'equation \eqref{eq:problemeglobal} discr\'etis\'ee avec un sch\'ema d'Euler s'\'ecrit sous la forme~:
-\begin{equation}
- \frac{\omega^{n+1}-\omega^{n}}{\Delta t} = \nabla u \omega^n
-\end{equation}
-cela nous donne le r\'esultat suivant~:
-\begin{equation}
- \omega^{n+1}=(1+\nabla u. \Delta t) \omega^n
-\end{equation}
-La condition pour le sch\'ema d'Euler est alors que $|1+X|<1$, avec $X=\nabla u.\Delta t$.
-\begin{figure}[h]
-\centering
- \includegraphics[width=0.6\linewidth]{Euler}
-\caption{Repr\'esentation graphique des \'equations $y=|1+x|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta t$}
-\end{figure}
-Il faut donc que $X= \nabla u.\Delta t$ soit inf\'erieure \`a $2$.\\
-On aura donc :
-\begin{center}
- $\nabla u. \Delta t <2$
-\end{center}
-En faisant le changement de variable, si la matrice $A=[\nabla u]$ est diagonalisable. Nous avons~:
-\begin{center}
- $\lambda_i . \Delta t <2$\hspace{0.2\linewidth} pour $i=0,1,2$\\
- $\Delta t < \frac{2}{\lambda_i}$
-\end{center}
-On utilise ensuite le th\'eor\`eme de Gerschgorin, qui permet de majorer les valeurs propres de la matrice $A$. On \'economise ainsi le nombre d'op\'eration \`a faire afin de d\'eterminer une condition pour le pas de temps $\Delta t$.
-\begin{Theoreme}[th\'eor\`eme de Gerschgorin\label{thm:gg}]
-Soit $A$ une matrice complexe de taille $n\times n$, de terme g\'en\'eral $a_{ij}$. Toute valeur propre de $A$ appartient au moins \`a un disque de Gerschgorin $\displaystyle D_i= \{z\in \mathbb{C}, |a_{ii}-z|\leq \sum_{i\neq j} |a_{ij}| \}$.
-\end{Theoreme}
-On peut donc majorer le rayon spectral de la matrice $A$ par $\displaystyle\max_i(\sum_{j} |a_{ij}|)$. On obtient ainsi la condition de stabilit\'e suivante pour le sch\'ema d'euler~:
-\begin{equation}
- \Delta t < \dfrac{2}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
-\end{equation}
-
- \item Le sch\'ema de Runge Kutta (ordre 2)
-Pour Runge-Kutta d'ordre 2, on obtient une majoration identique \`a celle d'Euler, car on a~:
-\begin{equation}
- \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 \bigr)\ \omega^{n}
-\end{equation}
-\begin{figure}[ht]
-\centering
- \includegraphics[width=0.6\linewidth]{RK2}
-\caption{Repr\'esentation graphique des \'equations $y=|1+x + \frac{1}{2}x^2|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta t$}
-\end{figure}
-
-On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
-\begin{equation}
- \Delta t < \dfrac{2}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
-\end{equation}
-
-
- \item Le sch\'ema de Runge Kutta (ordre 3)
-Pour Runge-Kutta d'ordre 3, on a~:
-\begin{equation}
- \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 + \frac{1}{6}(\nabla u.\Delta t)^3 \bigr)\ \omega^{n}
-\end{equation}
-\begin{figure}[ht]
-\centering
- \includegraphics[width=0.6\linewidth]{RK3}
-\caption{Repr\'esentation graphique des \'equations $y=|1+x+ \frac{1}{2}x^2+ \frac{1}{6}x^3|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta t$}
-\end{figure}
-
-On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
-\begin{equation}
- \Delta t < \dfrac{2.5127}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
-\end{equation}
-
- \item Le sch\'ema de Runge Kutta (ordre 4)
-Pour Runge-Kutta d'ordre 4, on a~:
-\begin{equation}
- \omega^{n+1} = \bigl(1 + \nabla u.\Delta t + \frac{1}{2}(\nabla u.\Delta t)^2 + \frac{1}{6}(\nabla u.\Delta t)^3 + \frac{1}{24}(\nabla u.\Delta)^4 \bigr)\ \omega^{n}
-\end{equation}
-\begin{figure}[ht]
-\centering
- \includegraphics[width=0.6\linewidth]{RK4}
-\caption{Repr\'esentation graphique des \'equations $y=|1+x+ \frac{1}{2}x^2+ \frac{1}{6}x^3+ \frac{1}{24}x^4|$ et $y = 1$, pour estimer les conditions \`a mettre sur $\Delta t$}
-\end{figure}
-
-On obtient alors en appliquant aussi le th\'eor\`eme~\ref{thm:gg}, la condition suivante pour le pas de temps~:
-\begin{equation}
- \Delta t < \dfrac{2.7853}{\displaystyle\max_i(\sum_{j} |\frac{\partial u_i}{\partial x_j}|)}
-\end{equation}
-\end{itemize}
-
-\begin{table}[ht]
-\centering
-\begin{tabular}{|c|c|c|}
-\hline
-Sch\'ema & $C_\text{Stretching}$ & Ordre \\
-\hline
-Euler & 2 &1 \\
-\hline
-Runge-Kutta 2($\alpha =\frac12$) & 2 & 2\\
-\hline
-Runge-Kutta 3 & 2.5127& 3 \\
-\hline
-Runge-Kutta 4 & 2.7853 & 4\\
-\hline
-\end{tabular}
-
-\caption{\label{tab:Cstretch} Tableau r\'ecapitulatif des constantes intervenants dans la condition de stabilit\'e du sch\'ema en temps}
-\end{table}
-
-
-Il faut maintenant comparer ces majorations \autoref{tab:Cstretch}, avec la condition de stabilit\'e sur l'advection qui est la suivante~:
-\begin{equation}
- \Delta t \simeq \dfrac{C}{\max_i|\frac{\partial u_i}{\partial x_i}|_{L_{\infty}}}
-\end{equation}
-avec $C$ qui vaut en g\'en\'eral 1.
-
-\section{\'Etude sur un cas test}
-\subsection{Cas test Analytique}
-L'\'etude de l'influence de la discr\'etisation en temps a \'et\'e faite sur l'exemple suivant~:
-\begin{align}
-\label{eq:exemplestretch}
- \overrightarrow{u} =
-\left(\begin{array}{l}
-2 \sin^2(\pi x) \sin (2 \pi y) \sin(2 \pi z) \cos(\pi t) \\
-- \sin(2 \pi x) \sin^2 (\pi y) \sin(2 \pi z) \cos(\pi t) \\
-- \sin(2\pi x) \sin (2 \pi y) \sin^2(\pi z) \cos(\pi t)
-\end{array}\right)&&\\
- \overrightarrow{w} =
-\left(\begin{array}{l}
-2 \pi \cos(\pi t) \sin(2 \pi x) \biggl(- \cos(2 \pi y) \sin^2(\pi z) + \cos(2 \pi z)\sin^2(\pi y) \biggr) \\
-2 \pi \cos(\pi t) \sin(2 \pi y) \biggl(2 \cos(2 \pi z) \sin^2(\pi x) + \cos(2 \pi x)\sin^2(\pi z) \biggr) \\
-- 2 \pi \cos(\pi t) \sin(2 \pi z) \biggl( \cos(2 \pi x) \sin^2(\pi y) + \cos(2 \pi y)\sin^2(\pi x) \biggr)
-\end{array}\right)&&\\
-\text{div} (\overrightarrow{u}) = 0 &&
-\end{align}
-La vorticit\'e est initialis'ee en prenant le rotationnel de u~:
-\begin{equation}
-\overrightarrow{w}= \overrightarrow{\text{rot}}u
-\end{equation}
-
-Pour cette exemple, et en prenant l'\'equation \autoref{eq:gradstretch} avec un sch\'ema aux diff\'erences finis centr\'es d'ordre 4 en espace avec un stencile de 5, on obtient les conditions de stabilit\'es suivantes pour le sch\'ema en temps \autoref{tab:stabexemple}.
-
-\begin{table}[ht]
- \centering
-\begin{tabular}{|c|c|c|}
-\hline
-\multicolumn{3}{|c|}{$\Delta t_\text{advection} = 0.44331613$} \\
-\hline
-\hline
-Sch\'ema & $\max(\Delta t_\text{Stretching})$ & Ordre \\
-\hline
-Euler & 0.14994711& 1\\
-\hline
-Runge-Kutta 2 &0.14994711 & 2\\
-Runge-Kutta 3 &0.18838605 & 3\\
-Runge-Kutta 4 &0.20882384 & 4\\
-\hline
-Adams-Moulton 3 & 0.44984132 & 3 \\
-\hline
-\end{tabular}
-\caption{\label{tab:stabexemple} Conditions de stabilit\'e sur le pas de temps pour l'exemple \eqref{eq:exemplestretch}}
-\end{table}
-
-
-\subsection{Cas test avec des donn\'ees discr\`etes d'entr\'ees}
-Jean-Baptiste a fourni un champs de vitesse discret avec turbulences anisotropes et h\'et\'erog\`enes, en $128^3$. Nous avons test\'e les diff\'erentes conditions de stabilit\'e \autoref{tab:stabexemple2}. (\`a Reynolds 80)\\
-Les donn\'ees en entr\'ees sont stock\'ees dans les fichiers~:
-\begin{itemize}
- \item Fields\_sav0\_U.data
- \item Fields\_sav0\_V.data
- \item Fields\_sav0\_W.data
-\end{itemize}
-
-\begin{table}[ht]
- \centering
-\begin{tabular}{|c|c|c|}
-\hline
-\multicolumn{3}{|c|}{$\Delta t_\text{advection} = 0.106679$} \\
-\hline
-\hline
-Sch\'ema & $\max(\Delta t_\text{Stretching})$ & Ordre \\
-\hline
-Euler & 0.124258& 1\\
-\hline
-Runge-Kutta 2 &0.124258 & 2\\
-Runge-Kutta 3 &0.151127 & 3\\
-Runge-Kutta 4 &0.173049 & 4\\
-\hline
-\end{tabular}
-\caption{\label{tab:stabexemple2} Conditions de stabilit\'e sur le pas de temps pour un champ de vitesse discret}
-\end{table}
-
-
-\end{document}
diff --git a/docs/tex_sources/Vortex_ring.pdf b/docs/tex_sources/Vortex_ring.pdf
deleted file mode 100644
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index 16ad78d536d025ff564cb96e65caab3e3004c502..0000000000000000000000000000000000000000
--- a/docs/tex_sources/Vortex_ring.tex
+++ /dev/null
@@ -1,113 +0,0 @@
-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={Vorticity_ring},
- pdfauthor={Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={Vorticity, Ring},
- pdfsubject={Vorticity}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Setup pour test de l'anneau de vorticité}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-
-L'initialisation de ce test se fait avec un anneau défini par une gaussienne, comme présenté dans Shariff et al. (1994) :
-\begin{equation}
- \omega_{\theta}(\mathbf{x}, t=0)=\dfrac{\Gamma_0}{\pi \sigma^2}e^{-(s/\sigma)^2}
-\end{equation}
-
-où $\Gamma_0$ représente la circulation à l'état initial, $\sigma$ le rayon interne du tore et où $s^2=(z-z_c)^2 + (\rvert(x, y) − (x_c, y_c)\rvert - R^2)$, avec $(x_c,y_c,z_c)$ les coordonnées du centre du tore.
-$\theta$ correspond ici à la deuxième coordonnée torique (rappel : coordonnées toriques : $(\vec e_R, \vec e_{\theta}, \vec e_{\phi})$, cf Figure, gauche)
-
-Le passage des coordonnée toriques aux coordonnées cartésiennes de la vorticité se fait de la façon suivante (cf Figure, droite):
-\begin{equation}
- \vec e_{\theta} = -\vec e_x \ sin(\theta) + \vec e_y \ cos(\theta)
-\end{equation}
-
-ainsi :
-\begin{align}
- & \omega_x = - \omega_{\theta} \ sin(\theta) \\
- & \omega_y = \omega_{\theta} \ cos(\theta) \\
- & \omega_z = 0
-\end{align}
-
-Soit $\vec v = (v_x,v_y,v_z) = (x-x_c, y-y_c,0)$, on peut alors exprimer $cos(\theta)$ et $sin(\theta)$ en fonction de $\mathbf{x}$ :\\
-\begin{align}
- & cos(\theta) = \dfrac{v_x}{\| \vec v \|} \\
- & sin(\theta) = \dfrac{v_y}{\| \vec v \|}
-\end{align}
-
-Les coordonnées cartésiennes de la vorticité sont donc données par:
-\begin{align}
- & \omega_x = - \omega_{\theta} \ \dfrac{v_y}{\| \vec v \|} \\
- & \omega_y = \omega_{\theta} \dfrac{v_x}{\| \vec v \|} \\
- & \omega_z = 0
-\end{align}
-
-\begin{figure}[ht]
-\centering
- \includegraphics[width=1.0\linewidth]{tore.pdf}
-\caption{(gauche) Systèmes de coordonnées cartésiennes et toriques sur un anneau 3D, (droite) Systèmes de coordonnées cartésiennes et toriques dans le plan (xOy).}
-\label{tore}
-\end{figure}
-
-A intervalle de temps régulier, une perturbation de la position centrale de l'anneau est appliquée :
-\begin{align}
- & R'(\theta) = R_0 [1 +\xi f(\theta)] \\
- & f(\theta) = \sum_{n=1}^N A_n \text{sin}(n\theta) + B_n \text{cos}(n \theta) \\
- & A_n^2 + B_n^2 = 1
-\end{align}
-
-(source : Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime, P. J. Archer, T. G. Thomas, G. N. Coleman, J. Fluid Mech. (2008), vol. 598, pp. 201–226)
-
-\underline{Raisons de cette perturbation} : Le but ici est de générer des anneaux de vorticité très épais, et donc ayant une forte circulation. Ce type d'anneaux a en effet son importance dans des applications telles que le contrôle d'écoulement par jets synthétiques. Dans les dispositifs expérimentaux (Gharib et al. (1998)), il n'est pas possible d'atteindre des hautes circulations lorsque la couche limite est fine.
-Quand le maximum de la vitesse axiale atteint la vitesse de la couche limite cette dernière subit une force dirigée vers l'axe de symétrie; elle ne peut alors plus élargir l'anneau de vorticité principal puisque son intensité tend vers zero lorsque $\sigma=0$. Ainsi, en forçant la couche limite à s'éloigner de l'axe de symétrie (en augmentant $R$ en fonction du temps), il devient possible de générer plus de circulation dans l'anneau.
-
-(source : paragraphe 4.4 dans Numerical experiments on vortex ring formation, K. Mohseni, H. Ran and T. Colonius, J. Fluid Mech. (2001), vol. 430, pp. 267–282)
-
-Afin d'assurer la divergence nulle de la vorticité, une projection de Helmholtz est effectuée après chaque perturbation de $R$. (cf documentation sur la projection de Helmholtz)
-
-
-\underline{Paramètres numériques choisis pour les simulations} :
-
-
-$\Omega$ = $[0,6] \times [0,6] \times [0,6]$\\
-$(x_c,y_c,z_c)$ = $(3,3,3)$\\
-resolution = $128^3$ ou $256^3$ \\
-pour $Re_{\Gamma} = 750$, $\Gamma_0$=0.0075 et $\nu=10^-5$\\
-pour $Re_{\Gamma} = 7500$, $\Gamma_0$=0.0075 et $\nu=10^-6$\\
-$R_0=3\sigma$=1.5\\
-$\xi=0.01$ (amplitude de la perturbation du centre de l'anneau)
-
-
-
-
-\end{document}
diff --git a/docs/tex_sources/advection.tex b/docs/tex_sources/advection.tex
deleted file mode 100644
index b55e1fce65e0dc0b8f4e1ac1c885627f6d7aae51..0000000000000000000000000000000000000000
--- a/docs/tex_sources/advection.tex
+++ /dev/null
@@ -1,234 +0,0 @@
-
-\section{Advection et remaillage}
-\label{sec:advect-et-rema}
-
-L'advection est implémentée de trois manières différentes dans ParMeS. En fortran, à travers le module \texttt{scales2py}, en python et en OpenCL. Les détails de l'implémentation OpenCL sur GPU sont donnés en section~\ref{sec:GPUdetails}.
-
-Les versions fortran et OpenCL permetent de réaliser une advection multi-échelle. La vitesse étant connue sur une grille plus grossière que les quantités advectées. La vitesse de chaque particule est interpolée à partir de la grille grossière.
-
-
-\subsection{Formules de remaillage $\Lambda_{n,k}$}
-\label{sec:remaillage}
-
-
-Les formules de remaillage de la forme $\Lambda_{n,k}$ sont des fonctions polynomiales par morceaux, symétriques, de support $]{ -n/2-1, n/2-1 }[$, de régularité $C^k$ qui conservent les $n$ premiers moments. La formule est donnée par morceaux sur les intervalles entiers par un polynôme $p_i$ de degré $2k+1$, soit $n+2$ polynômes. Les différentes formules générées sont résumées dans le tableau~\ref{tab:formulesRemaillage}.
-
-
-
-
-Les contraintes sont les suivantes:
-\begin{itemize}
-\item Symétrie (seulement $n/2 + 1$ polynômes à chercher)
- \begin{equation}
- \label{eq:4}
- \Lambda_{n,k} (x) = \Lambda_{n,k} (-x)
- \end{equation}
- \begin{equation}
- \label{eq:7}
- \Lambda_{n,k} =
- \begin{cases}
- p_i(x) & \text{ si } |x| \in [n/2-i ; n/2+1-i [\\
- 0 & \text{ sinon}
- \end{cases}
-\end{equation}
- \begin{equation}
- \label{eq:6}
- p_i(x) = \sum_{j=0}^{2k+1}c_{i,j}x^j,\quad i=0,\dots , n/2 + 1
- \end{equation}
-\item Raccords $C^k$ entre les polynômes
-\item Conservation des moments discrets:
- \begin{equation}
- \label{eq:2}
- \sum_{l=-N}^N l^q\Lambda_{n,k}(s-l) = s^q,\quad 0<s<1, q=0\dots n
- \end{equation}
-\end{itemize}
-Ainsi, par ces contraintes, on obtient un système d'équations dont les inconnues sont les coefficients $c_{i,k}$. Le nombre d'équation est bien supérieur au nombre d'inconnues.
-
-
-Remarque 1: pour chaque formule obtenue, les moments continus sont conservés:
-\begin{equation}
- \label{eq:3}
- \int_{-\infty}^\infty x^q\Lambda_{n,k}(x) \mathrm{d}x =
- \begin{cases}
- 1 & \text{si } q = 0\\
- 0 & \text{sinon }
- \end{cases}
-\end{equation}
-
-Remarque 2: on obtient une formule exacte aux points de coordonnées entières:
- \begin{equation}
- \label{eq:5}
- \Lambda_{n,k} (i) = \begin{cases}
- 1 & \text{ si } i=0\\
- 0 & \text{sinon}
- \end{cases},\quad i \in \lbrace -n/2+1,\dots, n/2-1 \rbrace
- \end{equation}
-
-
-\begin{equation}
- \label{eq:lambda21}
- M'_4 = \Lambda_{2,1}(x) =
- \begin{cases}
- 1-{5 \over 2}|x|^2+{3 \over 2}|x|^3 & 0\leqslant |x|<1\\
- 2-4|x|+{5 \over 2}|x|^2-{1 \over 2}|x|^3 & 1\leqslant |x|<2\\
- 0 & |x| \geqslant 2
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:lambda22}
- \Lambda_{2,2}(x) =
- \begin{cases}
- 1-|x|^2-{9 \over 2}|x|^3+{15 \over 2}|x|^4-3|x|^5& 0\leqslant |x|<1\\
- -4+18|x|-29|x|^2+{43 \over 2}|x|^3-{15 \over 2}|x|^4+|x|^5& 1\leqslant |x|<2\\
- 0 & |x| \geqslant 2
- \end{cases}
-\end{equation}
-\begin{equation}
- \label{eq:lambda23}
- \Lambda_{2,3}(x) =
- \begin{cases}
- 1-x^2-15x^4+({75 \over 2})x^5-({63 \over 2})x^6+9x^7 & 0\leqslant |x|<1\\
- 32-168x+376x^2-460x^3+330x^4-({277 \over 2})x^5+({63 \over 2})x^6-3x^7 & 1\leqslant |x|<2\\
- 0 & |x| \geqslant 2
- \end{cases}
-\end{equation}
-\begin{equation}
- \label{eq:lambda24}
- \Lambda_{2,4}(x) =
- \begin{cases}
- 1-x^2-({105 \over 2})x^5+({357 \over 2})x^6-231x^7+135x^8-30x^9& 0\leqslant |x|<1\\
- -208+1432x-4304x^2+7420x^3-8085x^4+({11543 \over 2})x^5& \\ \hspace{3cm}-({5397 \over 2})x^6+797x^7-135x^8+10x^9& 1\leqslant |x|<2\\
- 0 & |x| \geqslant 2
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:lambda42}
- M'_6 = \Lambda_{4,2}(x) =
- \begin{cases}
- 1-{5 \over 4}|x|^2-{35 \over 12}|x|^3+{21 \over 4}|x|^4-{25 \over 12}|x|^5 & 0\leqslant |x|<1\\
- -4+{75 \over 4}|x|-{245 \over 8}|x|^2+{545 \over 24}|x|^3-{63 \over 8}|x|^4+{25 \over 24}|x|^5 & 1\leqslant |x|<2\\
- 18-{153 \over 4}|x|+{255 \over 8}|x|^2-{313 \over 24}|x|^3+{21 \over 8}|x|^4-{5 \over 24}|x|^5 & 2\leqslant |x|<3\\
- 0 & |x| \geqslant 3
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:lambda43}
- \Lambda_{4,3}(x) =
- \begin{cases}
- 1-{5 \over 4}|x|^2-{28 \over 3}|x|^4+{145 \over 6}|x|^5-{245 \over 12}|x|^6+{35 \over 6}|x|^7 & 0\leqslant |x|<1\\
- 31-{1945 \over 12}|x|+{2905 \over 8}|x|^2-{5345 \over 12}|x|^3+{1281 \over 4}|x|^4-{1615 \over 12}|x|^5& \\ \hspace{6cm}+{245 \over 8}|x|^6-{35 \over 12}|x|^7 & 1\leqslant |x|<2\\
- -297+{3501 \over 4}|x|-{8775 \over 8}|x|^2+{3029 \over 4}|x|^3-{3731 \over 12}|x|^4+{911 \over 12}|x|^5& \\ \hspace{6cm}-{245 \over 24}|x|^6+{7 \over 12}|x|^7 & 2\leqslant |x|<3\\
- 0 & |x| \geqslant 3
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:lambda44}
- \Lambda_{4,4}(x) =
- \begin{cases}
- 1-{5 \over 4}|x|^2+{1 \over 4}|x|^4-{100 \over 3}|x|^5+{455 \over 4}|x|^6-{295 \over 2}|x|^7+{345 \over 4}|x|^8-{115 \over 6}|x|^9 & 0\leqslant |x|<1\\
- -199+{5485 \over 4}|x|-{32975 \over 8}|x|^2+{28425 \over 4}|x|^3-{61953 \over 8}|x|^4+{33175 \over 6}|x|^5& \\ \hspace{4cm} -{20685 \over 8}|x|^6+{3055 \over 4}|x|^7-{1035 \over 8}|x|^8+{115 \over 12}|x|^9 & 1\leqslant |x|<2\\
- 5913-{89235 \over 4}|x|+{297585 \over 8}|x|^2-{143895 \over 4}|x|^3+{177871 \over 8}|x|^4-{54641 \over 6}|x|^5& \\ \hspace{4cm} +{19775 \over 8}|x|^6-{1715 \over 4}|x|^7+{345 \over 8}|x|^8-{23 \over 12}|x|^9 & 2\leqslant |x|<3\\
- 0 & |x| \geqslant 3
- \end{cases}
-\end{equation}
-
-
-\begin{equation}
- \label{eq:lambda63}
- \Lambda_{6,3}(x) =
- \begin{cases}
- 1-{49 \over 36}|x|^2-{959 \over 144}|x|^4+{2569 \over 144}|x|^5-{727 \over 48}|x|^6+{623 \over 144}|x|^7& 0\leqslant |x|<1\\
- {138\over 5}-{8617 \over 60}|x|+{12873 \over 40}|x|^2-{791 \over 2}|x|^3+{4557 \over 16}|x|^4-{9583 \over 80}|x|^5& \\ \hspace{7cm} +{2181 \over 80}|x|^6-{623 \over 240}|x|^7& 1\leqslant |x|<2\\
- -440+{25949 \over 20}|x|-{117131 \over 72}|x|^2+{2247 \over 2}|x|^3-{66437 \over 144}|x|^4+{81109 \over 720}|x|^5& \\ \hspace{7cm} -{727 \over 48}|x|^6+{623 \over 720}|x|^7& 2\leqslant |x|<3\\
- {3632\over 5}-{7456 \over 5}|x|+{58786 \over 45}|x|^2-633|x|^3+{26383 \over 144}|x|^4-{22807 \over 720}|x|^5& \\ \hspace{7cm} +{727 \over 240}|x|^6-{89 \over 720}|x|^7& 3\leqslant |x|<4\\
- 0 & |x| \geqslant 4
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:lambda64}
- \Lambda_{6,4}(x) =
- \begin{cases}
- 1-{49 \over 36}|x|^2+{7 \over 18}|x|^4-{3521 \over 144}|x|^5+{12029 \over 144}|x|^6-{15617 \over 144}|x|^7+{1015 \over 16}|x|^8-{1015 \over 72}|x|^9& 0\leqslant |x|<1\\
- -{877\over5}+{72583 \over 60}|x|-{145467 \over 40}|x|^2+{18809 \over 3}|x|^3-{54663 \over 8}|x|^4+{390327 \over 80}|x|^5& \\ \hspace{4cm} -{182549 \over 80}|x|^6+{161777 \over 240}|x|^7-{1827 \over 16}|x|^8+{203 \over 24}|x|^9& 1\leqslant |x|<2\\
- 8695-{656131 \over 20}|x|+{3938809 \over 72}|x|^2-{158725 \over 3}|x|^3+{2354569 \over 72}|x|^4-{9644621 \over 720}|x|^5& \\ \hspace{4cm} +{523589 \over 144}|x|^6-{454097 \over 720}|x|^7+{1015 \over 16}|x|^8-{203 \over 72}|x|^9& 2\leqslant |x|<3\\
- -{142528\over5}+{375344 \over 5}|x|-{3942344 \over 45}|x|^2+{178394 \over 3}|x|^3-{931315 \over 36}|x|^4+{5385983 \over 720}|x|^5& \\ \hspace{4cm} -{1035149 \over 720}|x|^6+{127511 \over 720}|x|^7-{203 \over 16}|x|^8+{29 \over 72}|x|^9& 3\leqslant |x|<4\\
- 0 & |x| \geqslant 4
- \end{cases}
-\end{equation}
-\begin{equation}
- \label{eq:lambda65}
- \Lambda_{6,5}(x) =
- \begin{cases}
- 1-{49 \over 36}|x|^2+{7 \over 18}|x|^4-{701 \over 8}|x|^6+{54803 \over 144}|x|^7-{32165 \over 48}|x|^8+{9555 \over 16}|x|^9-{38731 \over 144}|x|^{10}+{3521 \over 72}|x|^{11}& 0\leqslant |x|<1\\
- 1233-{617533 \over 60}|x|+{1544613 \over 40}|x|^2-{515179 \over 6}|x|^3+{502579 \over 4}|x|^4-{3809911 \over 30}|x|^5& \\ \hspace{2.5cm} +{3618099 \over 40}|x|^6-{10894163 \over 240}|x|^7+{251685 \over 16}|x|^8-{172123 \over 48}|x|^9+{38731 \over 80}|x|^{10}-{3521 \over 120}|x|^{11}& 1\leqslant |x|<2\\
- -181439+{16709441 \over 20}|x|-{125352311 \over 72}|x|^2+{13002493 \over 6}|x|^3-{64445353 \over 36}|x|^4+{30912301 \over 30}|x|^5& \\ \hspace{2.5cm}-{3373567 \over 8}|x|^6+{88345523 \over 720}|x|^7-{1194095 \over 48}|x|^8+{160657 \over 48}|x|^9-{38731 \over 144}|x|^{10}+{3521 \over 360}|x|^{11}& 2\leqslant |x|<3\\
- 1188352-{19108864 \over 5}|x|+{250837216 \over 45}|x|^2-{14600752 \over 3}|x|^3+{25437902 \over 9}|x|^4-{17195278 \over 15}|x|^5& \\ \hspace{2.5cm}+{13253241 \over 40}|x|^6-{49136309 \over 720}|x|^7+{471205 \over 48}|x|^8-{45083 \over 48}|x|^9+{38731 \over 720}|x|^{10}-{503 \over 360}|x|^{11}& 3\leqslant |x|<4\\
- 0 & |x| \geqslant 4
- \end{cases}
-\end{equation}
-\begin{equation}
- \label{eq:lambda66}
- \Lambda_{6,6}(x) =
- \begin{cases}
-1-{49 \over 36}|x|^2+{7 \over 18}|x|^4-{1 \over 36}|x|^6-{46109 \over 144}|x|^7+{81361 \over 48}|x|^8-{544705 \over 144}|x|^9+{655039 \over 144}|x|^{10}& \\ \hspace{9cm} -{223531 \over 72}|x|^{11}+{81991 \over 72}|x|^{12}-{6307 \over 36}|x|^{13} & 0\leqslant |x|<1\\
--{44291 \over 5}+{1745121 \over 20}|x|-{15711339 \over 40}|x|^2+{32087377 \over 30}|x|^3-{7860503 \over 4}|x|^4+{38576524 \over 15}|x|^5& \\ \hspace{2.5cm} -{24659323 \over 10}|x|^6+{84181657 \over 48}|x|^7-{74009313 \over 80}|x|^8+{17159513 \over 48}|x|^9-{7870247 \over 80}|x|^{10}& \\ \hspace{9cm} +{438263 \over 24}|x|^{11}-{81991 \over 40}|x|^{12}+{6307 \over 60}|x|^{13}& 1\leqslant |x|<2\\
-3905497-{424679647 \over 20}|x|+{3822627865 \over 72}|x|^2-{2424839767 \over 30}|x|^3+{3009271097 \over 36}|x|^4-{930168127 \over 15}|x|^5& \\ \hspace{2.5cm} +{305535494 \over 9}|x|^6-{9998313437 \over 720}|x|^7+{203720335 \over 48}|x|^8-{137843153 \over 144}|x|^9+{22300663 \over 144}|x|^{10}& \\ \hspace{9cm} -{6126883 \over 360}|x|^{11}+{81991 \over 72}|x|^{12}-{6307 \over 180}|x|^{13}& 2\leqslant |x|<3\\
--{255622144\over 5}+{971097344 \over 5}|x|-{15295867328 \over 45}|x|^2+{5442932656 \over 15}|x|^3-{2372571796 \over 9}|x|^4+{2064517469 \over 15}|x|^5& \\ \hspace{2.5cm} -{9563054381 \over 180}|x|^6+{2210666335 \over 144}|x|^7-{796980541 \over 240}|x|^8+{76474979 \over 144}|x|^9-{43946287 \over 720}|x|^{10}& \\ \hspace{9cm} +{343721 \over 72}|x|^{11}-{81991 \over 360}|x|^{12}+{901 \over 180}|x|^{13}& 3\leqslant |x|<4\\
- 0 & |x| \geqslant 4
- \end{cases}
-\end{equation}
-
-
-\begin{equation}
- \label{eq:lambda84}
- \Lambda_{8,4}(x) =
- \begin{cases}
- 1-{205 \over 144}x^2+{91 \over 192}x^4-{6181 \over 320}x^5+{6337 \over 96}x^6-{2745 \over 32}x^7+{28909 \over 576}x^8-{3569 \over 320}x^9& 0\leqslant |x|<1\\
- -154+{12757 \over 12}x-{230123 \over 72}x^2+{264481 \over 48}x^3-{576499 \over 96}x^4+{686147 \over 160}x^5& \\ \hspace{6cm}-{96277 \over 48}x^6+{14221 \over 24}x^7-{28909 \over 288}x^8+{3569 \over 480}x^9& 1\leqslant |x|<2\\
-{68776\over7}-{1038011 \over 28}x+{31157515 \over 504}x^2-{956669 \over 16}x^3+{3548009 \over 96}x^4-{2422263 \over 160}x^5& \\ \hspace{6cm}+{197255 \over 48}x^6-{19959 \over 28}x^7+{144545 \over 2016}x^8-{3569 \over 1120}x^9& 2\leqslant |x|<3\\
- -56375+{8314091 \over 56}x-{49901303 \over 288}x^2+{3763529 \over 32}x^3-{19648027 \over 384}x^4+{9469163 \over 640}x^5& \\ \hspace{6cm}-{545977 \over 192}x^6+{156927 \over 448}x^7-{28909 \over 1152}x^8+{3569 \over 4480}x^9& 3\leqslant |x|<4\\
- {439375\over7}-{64188125 \over 504}x+{231125375 \over 2016}x^2-{17306975 \over 288}x^3+{7761805 \over 384}x^4-{2895587 \over 640}x^5& \\ \hspace{6cm}+{129391 \over 192}x^6-{259715 \over 4032}x^7+{28909 \over 8064}x^8-{3569 \over 40320}x^9& 4\leqslant |x|<5\\
- 0 & |x| \geqslant 5
- \end{cases}
-\end{equation}
-
-\begin{equation}
- \label{eq:m8prime}
- M'_8(x) =
- \begin{cases}
-{5 \over 48}x^7-{11 \over 32}x^6+{7 \over 8}x^4-{35 \over 24}x^2+{151 \over 168} & 0\leqslant |x|<1\\
-{103 \over 112}-{7 \over 30}x-{7 \over 16}x^2-{7 \over 3}x^3+{63 \over 16}x^4-{7 \over 3}x^5+{99 \over 160}x^6-{1 \over 16}x^7 & 1\leqslant |x|<2\\
--{139 \over 336}+{217 \over 30}x-{805 \over 48}x^2+{49 \over 3}x^3-{133 \over 16}x^4+{7 \over 3}x^5-{11 \over 32}x^6+{1 \over 48}x^7 & 2\leqslant |x|<3\\
-{128 \over 21}-{256 \over 15}x+{56 \over 3}x^2-{32 \over 3}x^3+{7 \over 2}x^4-{2 \over 3}x^5+{11 \over 160}x^6-{1 \over 336}x^7& 3\leqslant |x|<4\\
- 0 & |x| \geqslant 4
- \end{cases}
-\end{equation}
-
-\paragraph{Résumé des formules disponibles}
-
-\begin{table}[h]\centering
-\begin{tabular}{@{}lccccccccc@{}} \toprule
- & & Moments & Régularité & Nb points & Degré & Support & \multicolumn{3}{c}{Implémentation} \\
- & & & & & & & Fortran & Python & OpenCL \\\midrule
-$\Lambda_{2,1}=M'_4$ &\eqref{eq:lambda21} & 2 & $C^1$ & 4 & 3 & $[-2;2]$ & \checkmark&\checkmark & \checkmark \\
-$\Lambda_{2,2}$ &\eqref{eq:lambda22} & 2 & $C^2$ & 4 & 5 & $[-2;2]$ & - &\checkmark & \checkmark\\
-$\Lambda_{2,3}$ &\eqref{eq:lambda23} & 2 & $C^3$ & 4 & 7 & $[-2;2]$ &- & \checkmark&\checkmark \\
-$\Lambda_{2,4}$ &\eqref{eq:lambda24} & 2 & $C^4$ & 4 & 9 & $[-2;2]$ &- & \checkmark& \checkmark\\\midrule
-$\Lambda_{4,2} = M'_6$ &\eqref{eq:lambda42} & 4 & $C^2$ & 6 & 5 & $[-3;3]$ &\checkmark & \checkmark&\checkmark\\
-$\Lambda_{4,3}$ &\eqref{eq:lambda43} & 4 & $C^3$ & 6 & 7 & $[-3;3]$ & -& \checkmark&\checkmark\\
-$\Lambda_{4,4}$ &\eqref{eq:lambda44} & 4 & $C^4$ & 6 & 9 & $[-3;3]$ & \checkmark &\checkmark & \checkmark\\\midrule
-$\Lambda_{6,3}$ &\eqref{eq:lambda63} & 6 & $C^3$ & 8 & 7 & $[-4;4]$ &- & \checkmark&\checkmark\\
-$\Lambda_{6,4}$ &\eqref{eq:lambda64} & 6 & $C^4$ & 8 & 9 & $[-4;4]$ & \checkmark &\checkmark &\checkmark\\
-$\Lambda_{6,5}$ &\eqref{eq:lambda65} & 6 & $C^5$ & 8 & 11 & $[-4;4]$ & -& \checkmark&\checkmark\\
-$\Lambda_{6,6}$ &\eqref{eq:lambda66} & 6 & $C^6$ & 8 & 13 & $[-4;4]$ &\checkmark & \checkmark&\checkmark \\\midrule
-$\Lambda_{8,4}$ &\eqref{eq:lambda84} & 8 & $C^4$ & 10 & 9 & $[-5;5]$ &\checkmark &\checkmark & \checkmark \\\midrule
-$M'_8$ & \eqref{eq:m8prime} & & $C^4$ & 8 & 7 & $[-4;4]$ &\checkmark & \checkmark&\checkmark \\ \bottomrule
- \end{tabular}
- \caption{Formules de remaillage disponibles.}
-\label{tab:formulesRemaillage}
-\end{table}
diff --git a/docs/tex_sources/energy_enstrophy_decay.pdf b/docs/tex_sources/energy_enstrophy_decay.pdf
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-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={Energy_Decay},
- pdfauthor={Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={Energy, Decay},
- pdfsubject={Energies computation and effective viscosity}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
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-%breaklinks=true,
-% pdfpagelabels=true
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-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Utilisation de la relation de décroissance de l'énergie cinétique et de l'enstrophie pour valider l'implémentation de l'opérateur de diffusion}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-La définition générale de l'enstrophie $E(u)$ d'un écoulement fluide dans une région $\Omega \subseteq \mathbb{R}^3$ est :
-\begin{equation}
- E(u)=\int_{\Omega} \| \nabla u \|^2 \ \mathrm dx
-\end{equation}
-
-On considère les équations de Navier-Stokes pour un écoulement visqueux, incompressible et homogène, sans force extérieure :
-\begin{align}
-\dfrac{\partial u}{\partial t} + (u \cdot \nabla)u + \nabla p -\nu \Delta u=0 \\
-\nabla \cdot u =0
-\end{align}
-
-Avec une densité normalisée à 1, l'énergie $e(u)$ est donnée par :
-\begin{equation}
- e(u)=\int_{\Omega} \| u \|^2 \ \mathrm dx
-\end{equation}
-
-Ici, on considère que l'énergie de l'écoulement fluide est réduite à l'énergie cinétique. On ne décrit pas l'énergie potentielle ou thermique.
-
-Si la condition d'adhérence est respectée et si les intégrales convergent, on peut utiliser la condition d'incompressibilié pour effectuer une intégration par partie dans la définition de l'enstrophie donnée précédemment :
-\begin{equation}
- E(u)=\int_{\Omega} \| \nabla u \|^2 \ \mathrm dx
-= \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx - \int_{\Omega} \| u \| \| \Delta u \| \ \mathrm dx
-\end{equation}
-
-or en utilisant la condition d'incompressibilité on obtient l'équation de Poisson suivante : $ \Delta u = - \nabla \times \omega$ et ainsi :
-\begin{equation}
- \int_{\Omega} \| u \| \| \Delta u \| \ \mathrm dx = \int_{\Omega} \| u \| \| \nabla \times \omega \| \ \mathrm dx
- = \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx - \int_{\Omega} \| \omega \|^2 \ \mathrm dx
-\end{equation}
-
-On a donc :
-\begin{equation}
- E(u)=\int_{\Omega} \| \nabla u \|^2 \ \mathrm dx =
- \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx - \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx + \int_{\Omega} \| \omega \|^2 \ \mathrm dx
-\end{equation}
-
-or :
-\begin{align}
- \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx &= \int_{\Omega} \| u \|^2 \ \mathrm dx - \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx \\
-\Longleftrightarrow \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx &= \dfrac{1}{2} \int_{\Omega} \| u \|^2 \ \mathrm dx = e(u)
-\end{align}
-
-et :
-\begin{align}
- \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx &= \int_{\Omega} \| u \|^2 \ \mathrm dx - \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx \\
-\Longleftrightarrow \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx &= \dfrac{1}{2} \int_{\Omega} \| u \|^2 \ \mathrm dx = e(u)
-\end{align}
-
-ainsi :
-\begin{align}
- E(u) &= e(u) - e(u) + \int_{\Omega} \| \omega \|^2 \ \mathrm dx \\
-\Longleftrightarrow E(u) &= \int_{\Omega} \| \omega \|^2 \ \mathrm dx
-\end{align}
-
-On suppose que l'on a une décroissance suffisante de la pression $p$ et de la vitesse $u$ à l'infini. On a alors :
-\begin{equation}
-\dfrac{\partial u}{\partial t} -\nu \Delta u=0
-\end{equation}
-
-En prenant le produit scalaire avec $\dfrac{1}{2} u$ de cette équation et en l'intégrant sur $\Omega$ on obtient :
-\begin{align}
- \dfrac{1}{2} \dfrac{\partial}{\partial t} \int_{\Omega} \| u \|^2 \ \mathrm dx - \nu \int_{\Omega} \| \Delta u \| \| u \| \ \mathrm dx = 0
-\end{align}
-
-or
-\begin{equation}
- \int_{\Omega} \| \Delta u \| \| u \| \ \mathrm dx =
- \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx - \int_{\Omega} \| \omega \|^2 \ \mathrm dx
-\end{equation}
-
-avec $ \int_{\Omega} \| u \| \| \omega \| \ \mathrm dx = \int_{\Omega} \| \nabla u \| \| u \| \ \mathrm dx \rightarrow 0$ d'après les hypothèses
-
-ainsi :
-\begin{equation}
- \int_{\Omega} \| \Delta u \| \| u \| \ \mathrm dx = - \int_{\Omega} \| \omega \|^2 \ \mathrm dx = - E(u)
-\end{equation}
-
-On obtient donc que l'énergie décroit proportionnellement à l'enstrophie :
-\begin{align}
- \dfrac{\partial}{\partial t} \ e(u) &= - \nu \ E(u)\\
-\Longleftrightarrow \ \ \ \dfrac{1}{2} \dfrac{\partial}{\partial t} \int_{\Omega} \| u \|^2 \ \mathrm dx &= - \nu \int_{\Omega} \| \omega \|^2 \ \mathrm dx
-\end{align}
-
-Cette relation de proportionnalité peut-être utilisée pour valider l'implémentation de l'opérateur de diffusion dans les équations de Navier-Stokes. En effet, la viscosité effective du fluide $\nu_{\text{eff}}$ doit être retrouvée en calculant :
-\begin{equation}
--\dfrac{1}{2} \ \ \dfrac{\dfrac{\partial}{\partial t} \int_{\Omega} \| u \|^2 \ \mathrm dx}{\int_{\Omega} \| \omega \|^2 \ \mathrm dx}
-\end{equation}
-
-
-
-
-\end{document}
diff --git a/docs/tex_sources/force_computation.tex b/docs/tex_sources/force_computation.tex
deleted file mode 100644
index aee3bf192ea053493b164acb70ce43dea3bc1dbf..0000000000000000000000000000000000000000
--- a/docs/tex_sources/force_computation.tex
+++ /dev/null
@@ -1,233 +0,0 @@
-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
- \usepackage{cancel}
-\usepackage{xcolor}
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={Force computation},
- pdfauthor={Mimeau Chloe},
- pdfcreator={PARMES},
- pdfkeywords={force, pressure, friction, drag, vorticity flux},
- pdfsubject={Force computation}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Calcul des forces exercées par un fluide sur un corps}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-Le vecteur de la force $\mathbf{F}$ exercée par un fluide sur un corps est donné par la formule suivante :
-\begin{equation}
-\mathbf{F} = - \dfrac{d}{dt} \int_{B'} \mathbf{u} \ d\textbf{x}
-\end{equation}
-%
-où $B'$ dénote le domaine autour du corps $B$.
-%
-En intégrant l'équation de Navier-Stokes sur $B'$ on obtient:
-\begin{align}
-\dfrac{d}{dt} \int_{B'} \mathbf{u} \ d\textbf{x} - \nu \int_{B'} \Delta\mathbf{u} \ d\textbf{x} + \int_{B'} \nabla p \ d\textbf{x} = 0 \\
-\Longleftrightarrow \mathbf{F} = - \nu \int_{B'} \Delta\mathbf{u} \ d\textbf{x} + \int_{B'} \nabla p \ d\textbf{x}
-%\Longleftrightarrow \bold{F} = \nu \int_{B'} \nabla \times \boldsymbol{\omega} \ d\textbf{x} + \int_{B'} \nabla p \ d\textbf{x}
-\end{align}
-%
-Cette expression suppose que la vorticité s'annule à l'infini, ce qui en pratique est toujours le cas.\\
-%
-Or
-\begin{equation}
-\int_{B'} \nabla p \ d\textbf{x} = \int_{\partial B} p \cdot \mathbf{n} \ d\text{s}
-\end{equation}
-%
-et
-\begin{equation}\label{friction}
--\nu \int_{B'} \Delta\mathbf{u} \ d\textbf{x} = -\nu \int_{\partial B} \boldsymbol{\omega} \times \mathbf{n} \ d\text{s}
-\end{equation}
-%
-en effet,
-\begin{equation}
--\nu \int_{B'} \Delta\mathbf{u} \ d\textbf{x} = -\nu \int_{B'} \nabla \cdot \nabla\mathbf{u} \ d\textbf{x} = -\nu \int_{\partial B} \nabla\mathbf{u} \cdot \mathbf{n} \ d\text{s}
-\end{equation}
-
-avec, en 2D
-\begin{equation}
-\nabla\mathbf{u} \cdot \mathbf{n}=\begin{pmatrix}
-\textcolor{red}{n_1 \ \partial_x u} + n_2 \ \partial_y u \\
-n_1 \ \partial_x v + \textcolor{blue}{n_2 \ \partial_y v}\end{pmatrix}
-\end{equation}
-
-et
-\begin{equation}
-\omega \times \mathbf{n}=\begin{pmatrix}
-\textcolor{red}{-n_2 \ \partial_x v} + n_2 \ \partial_y u \\
-n_1 \ \partial_x v \ \textcolor{blue}{ - n_1 \ \partial_y u}\end{pmatrix}
-\end{equation}
-
-or, d'après la condition d'incompressibilité on a : $\partial_x u = -\partial_y v$,
-%
-donc
-\begin{equation}
-\textcolor{red}{n_1 \ \partial_x u} = -n_1 \ \partial_y v = \textcolor{red}{-n_2 \ \partial_x v}
-\end{equation}
-%
-En procédant de même, on obtient aussi l'égalité suivante $\textcolor{blue}{n_2 \ \partial_y v = -n_1 \ \partial_y u}$. Par conséquent $\nabla\mathbf{u} \cdot \mathbf{n}=\omega \times \mathbf{n}$ et donc l'équation \ref{friction} est vérifiée. La démonstration s'effectue de façon analogue en 3D.
-
-\underline{Remarque} : L'équation \ref{friction} peut également être obtenue directement en appliquant la formule de Green :
-\begin{equation}
--\nu \int_{B'} \Delta \mathbf{u} \ d\textbf{x} = \nu \int_{B'} \nabla \times \boldsymbol{\omega} \ d\textbf{x} = -\nu \int_{\partial B} \boldsymbol{\omega} \times \mathbf{n} \ d\text{s}
-\end{equation}
-
-
-L'expression de $\mathbf{F}$ est donc donnée par :
-\begin{equation}
-\mathbf{F}= - \nu \int_{\partial B} \boldsymbol{\omega} \times \mathbf{n} \ d\text{s} + \int_{\partial B} p \cdot \mathbf{n} \ d\text{s}
-\end{equation}
-
-où la première intégrale représente la traînée de friction et la seconde désigne la traînée de pression.
-
-Nous détaillons maintenant l'expression de cette force dans le cas d'un écoulement fluide 2D autour d'un cylindre.
-
-\underline{Calcul de la traînée de pression $F_p=\int_{\partial B} p \cdot n \ d\text{s}$}
-
-Le vecteur normal unitaire sortant $n$ vérifie la relation suivante :
-\begin{equation}
-\dfrac{\partial \Gamma}{\partial s} = \kappa(s) n(s)
-\end{equation}
-où $\kappa(s)$ représente la courbure de l'arc $s$ de la surface considérée. Dans le cas d'un cylindre $\kappa(s)=\kappa=1/R$ où $R$ désigne le rayon du cylindre et donc $n=R \dfrac{\partial \Gamma}{\partial s}$.
-
-Ainsi, en remplaçant $n$ par cette expression puis en effectuant une intégration par partie, on obtient :
-\begin{equation}
-\int_{\partial B} p \cdot n \ d\text{s}= R \int_{\partial B} p \cdot \dfrac{\partial \Gamma}{\partial s} \ d\text{s} = -R \int_{\partial B} \dfrac{\partial p}{\partial s} \cdot \Gamma \ d\text{s}
-\end{equation}
-%
-On a alors :
-\begin{equation}
-\int_{\partial B} p \cdot n \ d\text{s}= -R \int_{\partial B} \dfrac{\partial p}{\partial s} \cdot \Gamma \ d\text{s}
-\end{equation}
-%
-D'après l'équation de la quantité de mouvement l'égalité suivante est vérifiée :
-\begin{equation}
-\nu (\nabla \times \omega) \cdot \Gamma + \dfrac{\partial p}{\partial s} \cdot \Gamma = 0
-\end{equation}
-%
-Ainsi,
-\begin{equation}
--R \nu \int_{\partial B} (\nabla \times \omega) \cdot \Gamma \ d\text{s} - R \int_{\partial B} \dfrac{\partial p}{\partial s} \cdot \Gamma \ d\text{s} = 0
-\end{equation}
-%
-Et par conséquent,
-\begin{align}
-\int_{\partial B} p \cdot n \ d\text{s} &= R \nu \int_{\partial B} (\nabla \times \omega) \cdot \Gamma \ d\text{s}
-%\Longleftrightarrow \int_{\partial B} p \cdot n \ d\text{s} &= R \nu \int_{\partial B} ( \partial_y \omega , -\partial_x \omega) \cdot \Gamma \ d\text{s}
-\end{align}
-
-L'objectif est ici de montrer que cette expression est égale à celle du flux de vorticité à la surface du cylindre, autrement dit que :
-\begin{equation}\label{pressure}
-\int_{\partial B} p \cdot n \ d\text{s} = - R \nu \int_{\partial B} \dfrac{\partial\omega}{\partial n} \ d\text{s}
-\end{equation}
-%
-or
-\begin{equation}
-(\nabla \times \omega) \cdot \Gamma = \Gamma_1 \ \partial_y \omega - \Gamma_2 \ \partial_x \omega
-\end{equation}
-%
-et
-\begin{equation}
--\dfrac{\partial\omega}{\partial n} = -\nabla \omega \cdot n = -n_1 \ \partial_x \omega - n_2 \ \partial_y \omega
-\end{equation}
-%
-Par conséquent l'équation \ref{pressure} est vraie si et seulement si $\Gamma_1=-n_2$ et $\Gamma_2=n_1$, ce qui dans le cas du cylindre 2D est vérifié. La traînée de pression s'exprime donc comme le flux de vorticité à la surface du cylindre, c'est à dire :
-%
-\begin{equation}
-\int_{\partial B} p \cdot n \ d\text{s} = \nu \int_{\partial B} \dfrac{\partial\omega}{\partial n} \cdot \Gamma \ d\text{s} = \nu \int_{\partial B} \dfrac{\partial\omega}{\partial n} e_{\theta} \ d\text{s} \ \textcolor{red}{=} \ -\nu R \int_{0}^{2\pi} \dfrac{\partial\omega}{\partial n} e_{\theta} \ d\theta
-\end{equation}
-%
-Dans le cas du cylindre 2D on a donc :
-\begin{equation}
-\boxed{F_p = -\nu R \int_{0}^{2\pi} \dfrac{\partial\omega}{\partial n} e_{\theta} \ d\theta}
-\end{equation}
-%
-Et dans ce cas, le coefficient de pression est donné par
-\begin{equation}
-C_p = \dfrac{F_p \cdot e_x}{u_{\infty}^2 \rho R} = -\dfrac{\nu}{u_{\infty}^2 \rho}\int_{0}^{2\pi} \dfrac{\partial\omega}{\partial n} e_{\theta} \ d\theta \ \cdot e_x
-\end{equation}
-%
-or
-\begin{equation}
-e_{\theta} \cdot e_x = -(e_x \cdot e_x) \text{sin} \theta + (e_y \cdot e_x)\text{cos} \theta = -\text{sin} \theta
-\end{equation}
-%
-donc
-\begin{equation}
-\boxed{C_p = \dfrac{\nu}{u_{\infty}^2 \rho}\int_{0}^{2\pi} \dfrac{\partial\omega}{\partial n} \ \text{sin} \theta \ d\theta}
-\end{equation}
-
-\underline{Calcul de la traînée de friction $F_f=- \nu \int_{\partial B} \omega \times n \ d\text{s}$}
-
-
-On a
-\begin{equation}
-\omega \times n = (-n_2 \ \omega, n_1 \ \omega)
-\end{equation}
-%
-et
-\begin{equation}
-\omega \cdot \Gamma = (\Gamma_1 \ \omega, \Gamma_2 \ \omega)
-\end{equation}
-%
-or
-\begin{equation}
-\Gamma_1=-n_2 \ \ \text{et} \ \ \Gamma_2=n_1
-\end{equation}
-%
-donc
-\begin{equation}
-- \nu \int_{\partial B} \omega \times n \ d\text{s} = - \nu \int_{\partial B} \omega \cdot \Gamma \ d\text{s} = - \nu \int_{\partial B} \omega e_{\theta} \ d\text{s} \ \textcolor{red}{=} \ \nu R \int_{0}^{2\pi} \omega e_{\theta} \ d\theta
-\end{equation}
-%
-Dans le cas du cylindre 2D on a donc :
-\begin{equation}
-\boxed{F_f = \nu R \int_{0}^{2\pi} \omega e_{\theta} \ d\theta}
-\end{equation}
-%
-Et dans ce cas, le coefficient de friction est donné par
-\begin{equation}
-C_f = \dfrac{F_f \cdot e_x}{u_{\infty}^2 \rho R} = \dfrac{\nu}{u_{\infty}^2 \rho}\int_{0}^{2\pi} \omega e_{\theta} \ d\theta \ \cdot e_x
-\end{equation}
-%
-donc
-\begin{equation}
-\boxed{C_f = -\dfrac{\nu}{u_{\infty}^2 \rho}\int_{0}^{2\pi} \omega \ \text{sin} \theta \ d\theta}
-\end{equation}
-
-\vspace{2cm}
-
-\begin{figure}[ht]
-\centering
- \includegraphics[width=0.5\linewidth]{images/pressure_friction_drag.png}
-\caption{Les forces de pression ($PdA$) et de friction ($\tau_w dA$) s'exerçant sur l'élément de surface $dA$ d'un corps bidimensionnel.}
-\label{tore}
-\end{figure}
-
-\end{document}
diff --git a/docs/tex_sources/manual.pdf b/docs/tex_sources/manual.pdf
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-\documentclass[a4paper,10pt]{article}
-\usepackage{ucs,amsmath,a4wide,natbib, amssymb}
-\usepackage[utf8x]{inputenc}
-\usepackage{fancyhdr}
-\usepackage{lastpage}
-\usepackage{minted}
-\fancyhead[C]{}
-\fancyhead[R]{\texttt{\thepage/\pageref{LastPage}}}
-\fancyfoot[L]{}
-\fancyfoot[C]{}
-\fancyfoot[R]{\texttt{file main.tex -- \isodayandtime}}
-\usepackage{multirow}
-\setlength{\parskip}{12pt}
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync}
- \usepackage{cancel}
-\usepackage{xcolor}
-\usepackage[left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
-\overfullrule 5pt
-\usepackage[left=3cm,right=3cm,top=2.5cm,bottom=2.5cm]{geometry}
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-
-\usepackage[pdftex]{hyperref}
-\hypersetup{
- pdftitle={ParMeS User Guide},
- pdfauthor={ParMeS Devel},
- pdfcreator={ParMeS},
- pdfkeywords={particular methods, GPU, MPI},
-% pdfsubject={}
-% pagebackref=false,
-%colorlinks=false,citecolor=red,
-%linkcolor=blue,
-%plainpages=false,
-%breaklinks=true,
-% pdfpagelabels=true
-}
-
-\usepackage{mathtools}
-
-\DeclarePairedDelimiter\abs{\lvert}{\rvert}%
-\DeclarePairedDelimiter\norm{\lVert}{\rVert}%
-% Swap the definition of \abs* and \norm*, so that \abs
-% and \norm resizes the size of the brackets, and the
-% starred version does not.
-\makeatletter
-\let\oldabs\abs
-\def\abs{\@ifstar{\oldabs}{\oldabs*}}
-%
-\let\oldnorm\norm
-\def\norm{\@ifstar{\oldnorm}{\oldnorm*}}
-\makeatother
-
-\title{ParMeS User Guide}
-\author{ParMeS Development team}
-
-\begin{document}
-\maketitle
-\definecolor{codebg}{rgb}{0.95,1.0,0.97}
-
-
-\part{Introduction}
-\part{Operators}
-\section{About operators}
-\input{operators}
-\section{Computational operators}
-\input{velocity_correction}
-\section{Monitors}
-%\input{monitors}
-\section{Redistribute operators}
-
-\input{advection}
-\end{document}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: t
-%%% End:
diff --git a/docs/tex_sources/monitors.tex b/docs/tex_sources/monitors.tex
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@@ -1,40 +0,0 @@
-
-\subsection{Introduction}
-\subsection{Printer}
-\subsection{DragAndLift}
-\subsection{Reprojection$\_$criterion}
-Purpose : compute a projection criterion, depending on the current vorticity value on the grid.
-
-IN : vorticity field, reproj$\_$cst\\
-OUT : reprojRate, projCriterion
-with\\
- reprojRate : how often (in terms of iterations) this criteria must be taken into account \\
- and checkCriterion : true if reprojRate must be checked and updated (if necessary).
-
-\begin{itemize}
-\item Case 1: checkCriterion == True.\\
-If checkCriterion is true, the operator computes a criterion projection every reprojRate iteration.
-If this criterion is greater than reproj$\_$cst (user defined constant), then reprojRate is set to 1.
-reprojRate is reset to defaultRate (user-defined) at the beginning of every apply.
-\item Case 2 : checkCriterion == False.\\
-The operator will do nothing but give the (user-defined) value for reprojRate.
-\end{itemize}
-
-If io$\_$params is set, then a file is appended with some diagnostics values (see below) each reprojRate iterations.\\
-
-$\Omega$ being the whole domain, $\omega$ the vorticity field, let us denote :
-\begin{eqnarray}
-d1 &=& \max_{\Omega}\abs{\sum_j \frac{\partial \omega_j}{\partial x_j}}\\
-d2 &=& \max_{\Omega}\abs{\frac{\partial \omega_i}{\partial x_j}} \ \ \forall i,j \in [0,2]
-\end{eqnarray}
-
-And so, diagnostics = simulation.time, projCriterion, d1, d2, counter \\
-
-counter being the number of projection done since the beginning of the simulation.
-
-
-\subsection{Energy$\_$enstrophy}
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "manual"
-%%% End:
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-% \documentclass[a4paper,10pt]{article}
-% \usepackage[utf8x]{inputenc}
-% \usepackage{amsmath}
-% % \usepackage{showkeys} % affiche les labels et references
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-% \usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
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-% \overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-%
-% \usepackage[pdftex]{hyperref}
-% \hypersetup{
-% pdftitle={Flow past 3D sphere},
-% pdfauthor={Mimeau Chloe},
-% pdfcreator={PARMES},
-% pdfkeywords={sphere, potential flow, velocity correction, flow rate},
-% pdfsubject={Flow past 3D sphere}
-% % pagebackref=false,
-% %colorlinks=false,citecolor=red,
-% %linkcolor=blue,
-% %plainpages=false,
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-% % pdfpagelabels=true
-% }
-%
-%
-% \theoremstyle{plain}
-% \newtheorem{Theoreme}{Th\'eor\`eme}[section]
-% %opening
-% \title{Correction de vitesse pour la modélisation d'un écoulement incompressible dans une boîte}
-% \author{}
-% \date{}
-%
-% \begin{document}
-%
-% \maketitle
-Lors de la résolution de l'équation de Poisson $\Delta \mathbf{u} = - \nabla \boldsymbol{\omega} $, le mode 0 des transformées de Fourier rapides de la vitesse est imposé à 0, ce qui entraîne une modification des valeurs moyennes des composantes de la vitesse et donc des composantes de la vitesse elles-mêmes lorsque la transformée de Fourier inverse est appliquée. Une correction de la vitesse est donc nécessaire après la résolution de l'équation de Poisson, à chaque itération.
-
-On introduit la notation suivante pour exprimer la vitesse corrigée :
-%
-\begin{align}
-\bf{u}= \widetilde{u} + \overline{u}\label{ux}
-\end{align}
-
-où $\widetilde{\bf{u}} $ désigne la vitesse en sortie de FFT et où $\overline{\bf{u}} $ dénote la moyenne en espace de la vitesse, que l'on cherche ici à déterminer. L'écoulement s'effectue dans la direction $x$ et la vitesse est initialisée en posant ${\bf{u}}(t=0)=(u_{x \infty}, u_{y \infty}, u_{z \infty})$.
-
-
-$\overline{u_x}$ est un flux constant en espace, par conséquent $\dfrac{\partial \overline{u_x}}{\partial x} = \dfrac{\partial \overline{u_x}}{\partial y} = \dfrac{\partial \overline{u_x}}{\partial z} = 0$. Ainsi le champ de vorticité moyen en espace est donné en 2D par :
-\begin{align}\label{vortMoy}
-\overline{\omega} & = \dfrac{\partial \overline{u_y}}{\partial x}
-\end{align}
-et en 3D par :
-\begin{align}
-\overline{\omega_x} & = \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z} \\
-\overline{\omega_y} & = -\dfrac{\partial \overline{u_z}}{\partial x} \label{wy} \\
-\overline{\omega_z} & = \dfrac{\partial \overline{u_y}}{\partial x} \label{wz}
-\end{align}
-
-La correction de la vitesse se fait par réajustement du débit à l'entrée du domaine de calcul. Cette correction est développée ci-dessous composante par composante tout d'abord en 2D puis sera étendue au cas 3D.
-
-%%%%%%%%%%% 2D %%%%%%%%%%%%%%
-{\bf{2D}}
-
-\underline{Correction de $u_x$}:\\
-
-On note $B$ le coté correspondant au bord d'entrée du domaine de calcul. En partant de la décomposition \ref{ux} on a alors :
-\begin{align}
-\underbrace{\int\limits_B u_x \ dy}_{\text{débit souhaité}} = \underbrace{\int\limits_B \widetilde{u_x} \ dy}_{\text{débit calculé}} + \int\limits_B \overline{u_x} \ dy \\
-~
-\Longleftrightarrow \underbrace{\int\limits_B u_x \ dy}_{\text{débit souhaité}} = \underbrace{\int\limits_B \widetilde{u_x} \ dy}_{\text{débit calculé}} + \ L_y \overline{u_x} \\ \label{debSouhaiteUx}
-~
-\Longleftrightarrow \ \ \overline{u_x} = \dfrac{\text{débit souhaité}}{L_y} - \dfrac{\text{débit calculé}}{L_y}
-\end{align}
-
-où $L_y$ dénote la longueur du coté $B$. Or le débit que l'on souhaite imposer à l'entrée du domaine est égal à $u_{x \infty} \ L_y$ et donc :
-\begin{equation}
-\boxed{u_x = u_{x \infty} + \widetilde{u_x} - \dfrac{\int\limits_B \widetilde{u_x} \ dy}{L_y}}
-\end{equation}
-
-
-
-\underline{Correction de $u_y$}:\\
-
-$\overline{u_y}$ est un flux invariant en $y$. \\
-En effet, la vitesse est a divergence nulle, donc la moyenne en espace de la vitesse l'est également et ainsi $\partial_x \overline{u_x} + \partial_y \overline{u_y} = 0$. Or $\overline{u_x}$ est un flux constant en espace, par conséquent on a bien $\partial_y \overline{u_y} = 0$ (invariance de $\overline{u_y}$ en $y$).\\
-
-$\overline{u_y}$ est donc un flux variant en $x$ et d'après la relation \ref{vortMoy} on a $\overline{u_y}= \overline{\omega} x + c$, avec $c$ une constante. Ainsi :
-\begin{align}
-\underbrace{\int\limits_B u_y \ dy}_{\text{débit souhaité}} = \underbrace{\int\limits_B \widetilde{u_y} \ dy}_{\text{débit calculé}} + \int\limits_B \overline{\omega} x \ dy + \int\limits_B c \ dy
-\end{align}
-
-Or ici le débit souhaité est égal a 0. Donc :
-\begin{align}
-0 = \underbrace{\int\limits_B \widetilde{u_y} \ dy}_{\text{débit calculé}} + \overline{\omega} x_0 L_y + L_y c \\
-~
-\Longleftrightarrow c = -\dfrac{\int\limits_B \widetilde{u_y} \ dy}{L_y} - \overline{\omega} x_0
-\end{align}
-
-et donc :
-\begin{equation}
-\boxed{u_y = u_{y \infty} + \widetilde{u_y} + \overline{\omega}(x-x_0) - \dfrac{\int\limits_B \widetilde{u_y} \ dy}{L_y}}
-\end{equation}
-
-où $x_0$ dénote la coordonnée dans la direction $x$ du bord d'entrée du domaine de calcul.
-
-%%%%%%%%%%% 3D %%%%%%%%%%%%%%
-{\bf{3D}}
-
-\underline{Correction de $u_x$}:\\
-
-On note $S$ la surface correspondant au bord d'entrée du domaine de calcul. En partant de la décomposition \ref{ux} on a alors :
-\begin{align}
-\underbrace{\iint\limits_S u_x \ dydz}_{\text{débit souhaité}} &= \underbrace{\iint\limits_S \widetilde{u_x} \ dydz}_{\text{débit calculé}} + \iint\limits_S \overline{u_x} \ dydz \\
-\Longleftrightarrow \ \ \overline{u_x} &= \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} \label{debSouhaiteUx3D} \nonumber \\
-\Longleftrightarrow \ \ \overline{u_x} &= u_{x \infty} - \dfrac{\iint\limits_S \widetilde{u_x} \ dydz}{L_yL_z} \label{debSouhaiteUx3D} \nonumber
-\end{align}
-
-et donc :
-\begin{equation}
-\boxed{u_x = u_{x \infty} + \widetilde{u_x} - \dfrac{\iint\limits_S \widetilde{u_x} \ dydz}{L_yL_z}}
-\end{equation}
-
-\underline{Correction de $u_y$}:\\
-
-D'après la relation \ref{wz} on a $\overline{u_y} = \overline{\omega_z}x +c_1$ avec $c_1$ une constante. D'après \ref{ux} on obtient donc que $u_y = \widetilde{u_y} + \overline{\omega_z} x +c_1$. Ainsi,
-\begin{align}
-\iint\limits_S u_y \ dydz = \iint\limits_S \widetilde{u_y} \ dydz + \iint\limits_S \overline{\omega_z}x \ dydz + \iint\limits_S c_1 \ dydz \\
-~
-\Longleftrightarrow \ \underbrace{\iint\limits_S u_y \ dydz}_{\text{débit souhaité}} = \underbrace{\iint\limits_S \widetilde{u_y} \ dydz}_{\text{débit calculé}} + \ \overline{\omega_z}\ x_0L_yLz + c_1 L_yL_z
-\end{align}
-
-où $x_0$ désigne la coordonnée dans la direction $x$ de la surface d'entrée $S$ du domaine de calcul.
-
-Donc
-\begin{align}
-c_1 &= \dfrac{\text{débit souhaité}}{L_yL_z} - \dfrac{\text{débit calculé}}{L_yL_z} - \overline{\omega_z}\ x_0 \nonumber \\
-c_1 &= u_{y \infty} - \dfrac{\iint\limits_S \widetilde{u_y} \ dydz}{L_yL_z} - \overline{\omega_z}\ x_0 \nonumber
-\end{align}
-
-Et ainsi :
-\begin{equation}
-\boxed{u_y = u_{y \infty} + \widetilde{u_y} + \overline{\omega_z}(x-x_0) - \dfrac{\iint\limits_S \widetilde{u_y} \ dydz}{L_yL_z}}\label{correc_Y}
-\end{equation}
-
-\underline{Correction de $u_z$}:\\
-
-Sur le même principe on a d'après la relation \ref{wy} que $\overline{u_z} = -\overline{\omega_y}x +c_2$ avec $c_2$ une constante. D'après \ref{ux} on obtient donc que $u_z = \widetilde{u_z} - \overline{\omega_y} x +c_2$. Ainsi en procédant de même que pour $u_y$ on obtient :
-\begin{equation}
-\boxed{u_z = u_{z \infty} + \widetilde{u_z} - \overline{\omega_y}(x-x_0) - \dfrac{\iint\limits_S \widetilde{u_z} \ dydz}{L_yL_z}}\label{correc_Z}
-\end{equation}
-
-%%%%%%%% Remarque 1 pour la constance des terme sde correction en Y et en Z %%%%%%%%%%
-
-{\bf{Remarque 1:}}
-
-Nous allons justifier ici la constance des termes $c_1$ et $c_2$ utilisés dans les corrections de $u_y$ et $u_z$.
-
-Tout d'abord, revenons sur la constante $\overline{\omega_x}$. Cette constante, de part les conditions de bord choisies dans notre problème, est nulle. En effet :
-\begin{align}
-\overline{\omega_x} &= \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z}\\
-~
-\Longleftrightarrow \ \iiint\limits_{\Omega} \overline{\omega_x} &= \iiint\limits_{\Omega} \dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z} =0 \label{wx_zero}
-\end{align}
-car, en effet, des conditions périodiques sont imposées sur les bords dans les directions $Y$ et $Z$. Par conséquent, $\overline{\omega_x}$ est nul.
-
-De façon générale, à partir des relations \ref{wy} et \ref{wz}, on peut exprimer les flux $\overline{u_x}$, $\overline{u_y}$ et $\overline{u_z}$ comme suit :
-\begin{align}
-\overline{u_x} &= c_0\\
-\overline{u_y} &= \overline{\omega_z} x + f(y,z)\label{uy}\\
-\overline{u_z} &= -\overline{\omega_y} x + g(y,z)\label{uz}
-\end{align}
-%
-où $c_0$ est une constante et où $f$ et $g$ sont des fonctions dépendant de $y$ et de $z$.
-Ainsi, la jacobienne de $\bf{\overline{u}}$ est donnée par :
-
-\[J_{\overline{u}}(x,y,z)=
-\begin{pmatrix}
-\vspace{0.3cm}
- \dfrac{\partial \overline{u_x}}{\partial x} & \dfrac{\partial \overline{u_x}}{\partial y} & \dfrac{\partial \overline{u_x}}{\partial z} \\
-\vspace{0.3cm}
- \dfrac{\partial \overline{u_y}}{\partial x} & \dfrac{\partial \overline{u_y}}{\partial y} & \dfrac{\partial \overline{u_y}}{\partial z} \\
- \dfrac{\partial \overline{u_z}}{\partial x} & \dfrac{\partial \overline{u_z}}{\partial y} & \dfrac{\partial \overline{u_z}}{\partial z}
-\end{pmatrix}=
-\begin{pmatrix}
-\vspace{0.3cm}
- 0 & 0 & 0 \\
- \overline{\omega_z} & \dfrac{\partial f}{\partial y} & \dfrac{\partial f}{\partial z} \\
-\vspace{0.3cm}
- -\overline{\omega_y} & \dfrac{\partial g}{\partial y} & \dfrac{\partial g}{\partial z}
-\end{pmatrix}
-\]
-
-Par conséquent, à partir de la condition d'incompressibilité, on a :
-\begin{equation}
-\text{div}\ {\bf{u}} = 0 \ \Longleftrightarrow \ \dfrac{\partial \overline{u_x}}{\partial x} + \dfrac{\partial \overline{u_y}}{\partial y} + \dfrac{\partial \overline{u_z}}{\partial z} = 0 \ \Longleftrightarrow \ \boxed{\dfrac{\partial f}{\partial y} + \dfrac{\partial g}{\partial z} = 0}\label{div_f_g}
-\end{equation}
-
-D'autre part, on sait d'après \ref{wx_zero} que $\dfrac{\partial \overline{u_z}}{\partial y} - \dfrac{\partial \overline{u_y}}{\partial z}=0$, donc :
-\begin{equation}
-\boxed{\dfrac{\partial g}{\partial y} - \dfrac{\partial f}{\partial z} = 0}\label{rot_f_g}
-\end{equation}
-
-A partir de ces deux équations, on obtient les conditions suivantes sur $f$ et $g$ :
-\begin{center}
- $\partial_y$(\ref{div_f_g}) - $\partial_z$(\ref{rot_f_g}) $\Longleftrightarrow$ $\Delta f =0$ \\
-$\partial_z$(\ref{div_f_g}) + $\partial_y$(\ref{rot_f_g}) $\Longleftrightarrow$ $\Delta g =0$
-\end{center}
-
-
-Par conséquent, les solutions du système composé des équations \ref{div_f_g} et \ref{rot_f_g} sont les fonctions $f$ et $g$ s'exprimant comme les combinaisons linéaires suivantes de $y$ et $z$ :
-\begin{equation}
- \left\{
- \begin{align}
-f(y,z) &= ay + bz + c_1\\
-g(y,z) &= by - az + c_2, \qquad \text{avec} \ a,b, c_1, c_2 \in \mathbb{R}\label{syst_f_g}
- \end{align}
- \right.
-\end{equation}
-
-\underline{But} : Montrer que $f(y,z)=c_1$ et que $g(y,z)=c_2$. Ainsi la correction des composantes $u_y$ et $u_z$ de la vitesse sera bien donnée par les équations \ref{correc_Y} et \ref{correc_Z}.
-
-Sachant que les conditions imposées sur les bords du domaine dans les directions $Y$ et $Z$ sont des conditions périodiques, on a :
-\begin{align}
-\overline{u_y}(x,y_0,z) &= \overline{u_y}(x,y_{\text{max}},z), \\
-\overline{u_y}(x,y,z_0) &= \overline{u_y}(x,y,z_{\text{max}})
-\end{align}
-%
-où $y_0, z_0$ et $y_{\text{max}}, z_{\text{max}}$ désignent respectivement les coordonnées minimales et maximales du domaine dans les directions $Y$ et $Z$. Donc d'après la relation \ref{uy} et d'après l'expression de $f$ donnée dans le système \ref{syst_f_g}, on obtient les égalités suivantes :
-\begin{align}
-\overline{\omega_z}x + ay_0 + bz + c_1 &= \overline{\omega_z}x + ay_{\text{max}} + bz + c_1, \\
-\overline{\omega_z}x + ay + bz_0 + c_1 &= \overline{\omega_z}x + ay + bz_{\text{max}} + c_1
-\end{align}
-$\Longleftrightarrow$
-\begin{align}
-a(y_0- y_{\text{max}})=0\\
-b(z_0- z_{\text{max}})=0
-\end{align}
-
-La longueur du domaine étant non nulle dans les directions $Y$ et $Z$, alors $a=b=0$, et donc $f(y,z)=c_1$ et $g(y,z)=c_2$. Bien-sûr, le même résultat est obtenu en considérant la relation \ref{uz} et l'expression de $g$ donnée dans le système \ref{syst_f_g}.
-
-%%%%%%%% Remarque 2 pour écoulements potentiels autour cylindre/sphère %%%%%%%%%%%
-{\bf{Remarque 2:}}
-
-Dans le cadre d'un écoulement autour d'un cylindre (2D) ou d'une sphère (3D), on peut également choisir l'écoulement potentiel comme condition initiale de l'écoulement.
-
-Par décomposition de Helmholtz on a:
-\begin{equation}
-\mathbf{u} = \nabla \times \psi + \nabla \phi
-\end{equation}
-où la fonction courant $\psi$ et le potentiel scalaire $\phi$ vérifient les propriétés suivantes :
-\begin{equation}
-\text{div} \ (\psi) = 0, \ \ \ \
-\nabla \times \nabla \phi= 0
-\end{equation}
-Donc si l'écoulement est potentiel, c'est à dire irrotationnel, alors :
-\begin{equation}
-\mathbf{u} = \nabla \phi
-\end{equation}
-et \underline{en 2D}, la fonction courant et le potentiel scalaire sont liés par la relation suivante :
-\begin{equation}
-\dfrac{\partial \phi}{\partial x} = \dfrac{\partial \psi}{\partial y}, \ \ \
-\dfrac{\partial \phi}{\partial y} = -\dfrac{\partial \psi}{\partial x}
-\end{equation}
-Ainsi $u_x = \dfrac{\partial \psi}{\partial y} $ et $u_y= -\dfrac{\partial \psi}{\partial x}$.
-
-La fonction courant associée à un écoulement potentiel 2D autour d'un cylindre de rayon $R$ est exprimée de la façon suivante :
-\begin{equation}
-\psi = u_{\infty} y \left ( 1 + \dfrac{R^2}{(x-x_C)^2 + (y-y_C)^2} \right )
-\end{equation}
-
-avec $(x_C, y_C)$ les coordonnées du centre du cylindre dans le repère cartésien du domaine de calcul.
-
-Ainsi l'expression du débit souhaité, intervenant dans l'équation \ref{debSouhaiteUx}, est désormais donnée par :
-\begin{equation}
-\underbrace{\int\limits_B u_x \ dy}_{\text{débit souhaité}} = \int\limits_B \dfrac{\partial \psi}{\partial y} \ dy = [\psi]_B = \left[ u_{\infty} y \left (1-\dfrac{R^2}{(x_0-x_C)^2-(y-y_C)^2}\right ) \right ]_B
-\end{equation}
-
-\underline{En 3D}, il est pratique d'exprimer le potentiel d'un écoulement autour d'une sphère de rayon $R$ dans un système de coordonnées sphériques $(r, \theta, \varphi)$, dans lequel l'angle dans la direction azimutale $\theta$ est mesuré à partir de la direction de l'écoulement. Dans ce système de coordonnées, le potentiel est indépendant de l'angle méridional $\varphi$ et est donné par :
-\begin{equation}
-\phi_{\text{sph}} = u_{\infty} r \ \text{cos} \theta\left ( 1 + \dfrac{R^3}{2r^3} \right )
-\end{equation}
-%
-Ainsi, les composantes de la vitesse en coordonnées sphériques sont les suivantes:
-\begin{align}
-\mathbf{u}_{\text{sph}} &= \nabla \phi_{\text{sph}} = (u_r, u_{\theta}, u_{\varphi}) = \left ( \dfrac{\partial \phi}{\partial r}, \ \dfrac{1}{r} \dfrac{\partial \phi}{\partial \theta}, \ \dfrac{1}{r\text{sin}(\theta)} \dfrac{\partial \phi}{\partial \varphi} \right) \\
-u_r &= \dfrac{\partial \phi}{\partial r} = u_{\infty} \text{cos} \theta \left ( 1 - \dfrac{R^3}{r^3} \right ) \\
-u_{\theta} &= \dfrac{1}{r}\dfrac{\partial \phi}{\partial \theta} = -u_{\infty} \text{sin} \theta \left ( 1 + \dfrac{R^3}{2 r^3} \right ) \\
-u_{\varphi} &= 0 \\
-\end{align}
-%
-Dans un système de coordonnées cartésiennes, le potentiel est donc donné par :
-\begin{equation}
-\phi_{\text{cart}} = u_{\infty} x \left ( 1 + \dfrac{R^3}{2r^3} \right )
-\end{equation}
-où $x=r \text{cos}\theta$ et $r=\sqrt{(x-x_S)^2 + (y-y_S)^2 + (z-z_S)^2}$, avec $(x_S, y_S, z_S)$ les coordonnées du centre de la sphère dans le repère cartésien du domaine de calcul.
-
-Les composantes de la vitesse en coordonnées cartésiennes sont donc :
-\begin{align}
-\mathbf{u}_{\text{cart}} &= \nabla \phi_{\text{cart}} = (u_x, u_y, u_z) \\
-u_x &= u_{\infty} \left ( 1 - \dfrac{3R^3x^2}{2r^5} + \dfrac{R^3}{2r^3}\right ) \\
-u_y &= -u_{\infty} \dfrac{3R^3 xy}{2 r^5} \\
-u_z &= -u_{\infty} \dfrac{3R^3 xz}{2 r^5}
-\end{align}
-
-
-% Ainsi l'expression du débit souhaité, intervenant dans l'équation \ref{debSouhaiteUx3D}, est désormais donnée par:
-% \begin{equation}
-% \underbrace{\iint\limits_S u_x \ dydz}_{\text{débit souhaité}} = \iint\limits_S \ \dfrac{-u_{\infty} x}{r} \left ( 1 - \dfrac{R^3}{r^3} \right ) \ dydz = -u_{\infty} x_0 \iint\limits_S \ \dfrac{1}{r} \left ( 1 - \dfrac{R^3}{r^3} \right ) \ dydz
-% \end{equation}
-
-% \end{document}
diff --git a/docs/tex_sources/vorticity_solenoidal_projection.pdf b/docs/tex_sources/vorticity_solenoidal_projection.pdf
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diff --git a/docs/tex_sources/vorticity_solenoidal_projection.tex b/docs/tex_sources/vorticity_solenoidal_projection.tex
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-\documentclass[a4paper,10pt]{article}
-\usepackage[utf8x]{inputenc}
-\usepackage{amsmath}
-% \usepackage{showkeys} % affiche les labels et references
-\usepackage[T1]{fontenc}
-\usepackage{multirow}
-\usepackage{amsthm,amssymb}
-\setlength{\parskip}{12pt} %taille du saut entre deux paragraphes
-\usepackage{graphicx}
-\graphicspath{{./images/}}
-\usepackage{pdfsync} % permet de cliquer sur le pdf pour avoir l endroit ou il y a le code latex
-\overfullrule 5pt %Pour d\'esigner les listes couples d'un *full carre noir la ou l on sort
-\usepackage[pdftex]{hyperref}
-
-\def\bmu{\mbox{\boldmath$\mu$}}
-\def\btau{\mbox{\boldmath$\tau$}}
-\def\bw{\mbox{\boldmath$\omega$}}
-\def\bpsi{\mbox{\boldmath$\psi$}}
-\def\bgamma{\mbox{\boldmath$\gamma$}}
-\def\bv{\mathbf{v}}
-\def\bu{\mathbf{u}}
-\def\bxi{\boldsymbol{\xi}}
-\def\obu{\overline{\mathbf{u}}}
-\def\obw{\overline{\bw}}
-
-
-\def\bnabla{\mbox{\boldmath$\nabla$}}
-\def\bDelta{\mbox{\boldmath$\Delta$}}
-\def\w{\mbox{$\omega$}}
-\def\btimes{\mbox{\boldmath$\times$}}
-\def\rot{\mbox{\boldmath$\nabla\times$}}
-\def\div{\bnabla\cdot}
-\def\cal#1{\mathcal{#1}}
-\def\Rr{\hbox{\rm I\hskip -1.5pt R}}
-\def\dpart#1#2{\frac{\partial#1}{\partial#2}}
-\def\dratio#1#2{\dfrac{#1}{#2}}
-\def\ds{\displaystyle}
-
-% \hypersetup{
-% pdftitle={Solenoidal_projection},
-% pdfauthor={Mimeau Chloe},
-% pdfcreator={PARMES},
-% pdfkeywords={vorticity, solenoidal, projection},
-% pdfsubject={Vorticity projection}
-% % pagebackref=false,
-% %colorlinks=false,citecolor=red,
-% %linkcolor=blue,
-% %plainpages=false,
-% %breaklinks=true,
-% % pdfpagelabels=true
-% }
-
-
-\theoremstyle{plain}
-\newtheorem{Theoreme}{Th\'eor\`eme}[section]
-%opening
-\title{Projection solenoidale du champ de vorticité par décomposition de Helmholtz en dimension 3}
-\author{}
-\date{}
-
-\begin{document}
-
-\maketitle
-
-L'algorithme de la méthode de projection est basé sur la décomposition de Helmholtz selon laquelle tout champ de vecteur peut se décomposer en deux parties : une partie solenoidale (ie, à divergence nulle) et une partie irrotationelle s'exprimant comme le gradient d'un potentiel scalaire :
-\begin{align}
- \boldsymbol{\omega} &= \boldsymbol{\omega_{\textbf{sol}}} + \boldsymbol{\omega_{\textbf{irr}}} = \boldsymbol{\omega_{\textbf{sol}}} + \nabla \boldsymbol{\pi} \\
- \nabla \cdot \boldsymbol{\omega_{\textbf{sol}}} &= 0 \\
- \nabla \times \boldsymbol{\omega_{\textbf{irr}}} &= 0
-\end{align}
-
-Ici, on applique la méthode de projection solenoidale au champ de vorticité $\omega$ afin de garantir la divergence nulle de ce champ tout au long de la résolution des équations de Navier-Stokes.
-L'algorithme se déroule en deux étapes. Tout d'abord, une vorticité intermédiaire non solénoidale $\boldsymbol{\tilde \omega}$ est calculée, puis dans un second temps le gradient du potentiel scalaire $\boldsymbol{\pi}$ est utilisé pour projeter la vorticité sur l'espace des champs de vorticité à divergence nulle. Le champ de vorticité à divergence nulle est donc obtenu en soustrayant à la vorticité intermédiaire sa partie irrotationnelle :
-\begin{align}
- \boldsymbol{\omega_{\textbf{sol}}} &= \boldsymbol{\tilde \omega} - \boldsymbol{\omega_{\textbf{irr}}} \\
- \Longleftrightarrow \boldsymbol{\omega_{\textbf{sol}}} &= \boldsymbol{\tilde \omega} - \nabla \boldsymbol{\pi}
-\end{align}
-
-La résolution de cette équation se fait en prenant sa divergence :
-\begin{align}
- \textbf{div}(\boldsymbol{\omega_{\textbf{sol}}}) &= \textbf{div}(\boldsymbol{\tilde \omega}) - \Delta \boldsymbol{\pi}
-\end{align}
-or $\textbf{div}(\boldsymbol{\omega_{\textbf{sol}}}) = 0$ donc :
-\begin{align}
- \textbf{div}(\boldsymbol{\tilde \omega}) - \Delta \boldsymbol{\pi} = 0
-\end{align}
-On obtient alors l'équation de Poisson suivante (pour des raisons de simplification d'écriture, la notation $\tilde \cdot$ a été supprimée):
-\begin{align}
-\Delta \boldsymbol{\pi} = \textbf{div}(\boldsymbol{\omega})
-\end{align}
-
-Pour résoudre cette équation on se place dans l'espace des fréquences en calculant sa transformée de Fourier :
-\begin{align}
-\widehat{\Delta \boldsymbol{\pi}} = \widehat{\textbf{div}(\boldsymbol{\omega})}
-\end{align}
-
-\begin{align}
-\widehat{\textbf{div}(\boldsymbol{\omega})} = \widehat{\left( \sum_{j=1}^3 \dfrac{\partial \boldsymbol{\omega_j}}{\partial x_j} \right)} = \sum_{j=1}^3 i \boldsymbol{\xi}_j \widehat{\boldsymbol{\omega}}_j
-\end{align}
-et
-\begin{align}
-\widehat{\Delta \boldsymbol{\pi}} = -\mid \boldsymbol{\xi} \mid^2 \widehat{\boldsymbol{\pi}}
-\end{align}
-
-Donc
-\begin{align}
--\mid \xi \mid^2 \widehat{\boldsymbol{\pi}} = \sum_{j=1}^3 i \boldsymbol{\xi}_j \widehat{\boldsymbol{\omega}}_j \\
-\Longleftrightarrow \widehat{\boldsymbol{\pi}} = - i \dfrac{\sum_{j=1}^3 \boldsymbol{\xi}_j \widehat{\boldsymbol{\omega}}_j}{\mid \boldsymbol{\xi} \mid^2}
-\end{align}
-
-D'après l'équation (5) on a :
-\begin{align}
-\widehat{\boldsymbol{\omega}'} &= \widehat{\boldsymbol{\omega_{\textbf{sol}}}} = \widehat{\boldsymbol{\omega}} - \widehat{\nabla \boldsymbol{\pi}} \\
-\Longleftrightarrow \widehat{\boldsymbol{\omega}'}_k &= \widehat{\boldsymbol{\omega}}_k - \widehat{\nabla \boldsymbol{\pi}}_k, \ 1\leq k \leq 3
-\end{align}
-
-or $\widehat{\nabla \boldsymbol{\pi}}_k = i \boldsymbol{\xi}_k \widehat{\boldsymbol{\pi}}_k \ $, ainsi on obtient :
-\begin{equation}
-\widehat{\boldsymbol{\omega}'}_k = \widehat{\boldsymbol{\omega}}_k - i \boldsymbol{\xi}_k \widehat{\boldsymbol{\pi}}_k \\
-\ \ \ \Longleftrightarrow
-\end{equation}
-\begin{equation}\label{proj}
-\boxed{\widehat{\boldsymbol{\omega}'}_k = \widehat{\boldsymbol{\omega}}_k - \boldsymbol{\xi}_k \dfrac{\sum_{j=1}^3 \boldsymbol{\xi}_j \widehat{\boldsymbol{\omega}}_j}{\mid \boldsymbol{\xi} \mid^2}}, \ 1\leq k \leq 3
-\end{equation}
-
-Un critère de reprojection peut-être donné par :
-\begin{equation}
-\max\limits_{x} \left ( \dfrac{\rvert \text{div} (\omega) \rvert }{\max\limits_{x} \left \rvert\dfrac{\partial \omega_i (x)}{\partial x_j} \right \rvert } \right ) \geq c
-\end{equation}
-
-avec $c$ une constante fixée inférieure à 1, de l'ordre de 1-10$\%$.
-
-\underline{Remarque} :
-
-Calculer le rotationnel de $\bu$ (c'est à dire calculer la vorticité à partir du champ de vitesse) revient à projeter la vorticité dans l'espace des champs à divergence nulle.
-
-\begin{proof}
-
-L'équation de Poisson sur la vitesse est donnée par :
-\begin{equation}
-\bDelta \bu = - \rot \bw
-\end{equation}
-ainsi, en passant dans l'espace des fréquences on obtient :
-\begin{equation}
-\widehat{\bu} = \dfrac{1}{\mid \bxi \mid^2} \rot \widehat{\bw}
-\end{equation}
-c'est à dire :
-\begin{equation}
-\widehat{\bu} = \dfrac{1}{\mid \bxi \mid^2} \ ( i \bxi \times \widehat{\bw})
-\end{equation}
-Par conséquent
-\begin{align}
-\widehat{\bw} &= \widehat{\rot \bu} \\
-&= i \bxi \times \widehat{\bu} \\
-&= i \bxi \times \left(\dfrac{1}{\mid \bxi \mid^2} \ ( i \bxi \times \widehat{\bw}) \right)
-\end{align}
-donc d'après l'égalité du double produit vectoriel
-\begin{equation}
-a \times (b \times c) = (a \cdot c)b - (a \cdot b)c,
-\end{equation}
-on a :
-\begin{align}
-\widehat{\bw} &= \widehat{\rot \bu} \\
-&= (i \bxi \cdot \widehat{\bw})\dfrac{1}{\mid \bxi \mid^2} \ i \bxi - \left(i \bxi \cdot \dfrac{1}{\mid \bxi \mid^2} \ i \bxi \right)\widehat{\bw}\\
-&= - \bxi \ \dfrac{(\bxi \cdot \widehat{\bw})}{\mid \bxi \mid^2} + \dfrac{\bxi^2}{\mid \bxi \mid^2} \ \widehat{\bw}\\
-&= - \bxi \ \dfrac{(\bxi \cdot \widehat{\bw})}{\mid \bxi \mid^2} + \widehat{\bw}\\
-&= \widehat{\bw_{\textbf{sol}}} \qquad \text{d'après l'équation \ref{proj}}
-\end{align}
-Par conséquent l'égalité suivante est vérifiée :
-\begin{equation}
-\boxed{\rot \bu = \bw = \bw_{\textbf{sol}}}
-\end{equation}
-
-\end{proof}
-
-
-
-\end{document}