.. _mesh_section:

Mesh
****

Introduction
============

The domain :math:`\mathcal{V}` is represented by a computationnal domain made of a representation of elementary volumes (cells) :math:`\mathcal{V}_i` such as \:

:math:`\mathcal{V}=\cup\mathcal{V}_i`.

.. important::
 Here, only  1D or 2D structured meshes are considered: the number of neighbours of a cell is constant (2 in 1D, 4 in 2D)  and the numbering of elementary volumes :math:`\mathcal{V}_i` is ordered.

1D Mesh
=======

In 1D, the cells are usually numbered with index *i*.

 .. figure:: ./_static/chapitre1_img/chapitre1_1.jpg
    :scale: 75%
    :align: left

    : 1D mesh of a domain :math:`\mathcal{V}` with its elementary volumes (cells) :math:`\mathcal{V}_i` 


In 1D,  to obtain a **structured** mesh one must have \:

 * The number of cells and the chosen numbering  (index values of the 1rst and last cell): usually *i* is used for the *x* direction.
 * The value of the volume, this is the length of each cell (constant or not) usually nammed :math:`\Delta x`

2D mesh
=======

.. figure:: ./_static/chapitre1_img/chapitre1_2.jpg
    :scale: 75%
    :align: left

    : 2D Mesh of a domain :math:`\mathcal{V}` in elementary volume :math:`\mathcal{V}_i`


In 2D, to obtain a **structured** mesh one must have:

 * The number of cells and the chosen numbering  (index values of the 1rst and last cell in each direction): generally *i* is used for the *x* direction and *j* for the transverse direction.
 * The value of the volume, this is the area of each cell, the surface of the boundary of the cell (face), this is the length of each of the 4 faces of the cell, and the exiting normal to every face :math:`\vec{n}_k`  , *k* being the number of the face, *k* is varying from 1 to 4. 

.. _figure_2Dcell:

.. figure:: ./_static/chapitre1_img/chapitre1_3.jpg
    :scale: 100%
    :align: center

    General quadrangle cell of a 2D mesh 



.. warning::
   For the code development (from 1D to 2D) the numbering of the faces is the same as the one depicted in  :numref:`figure_2Dcell`. Bottom face is number 1, then conterclockwise numbering.

Examples
========

.. toctree:: 
  
   chap2_mesh_exemple