Modélisation A
==============


Caractéristiques de la modélisation
-----------------------------------

La cellule est homogénéisée en MASSIF afin d’obtenir les paramètres pour un modèle 3D.


Valeurs testées
---------------

On compare les valeurs des coefficients homogénéisés (élastiques et thermiques) obtenues par le calcul à ceux de la note [Voldoire1993]_.


+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| Coefficients homogénéisés      | Référence                        | Calcul ASTER | Erreur relative (%)                                                                                                       |
|                                |                                  |              | :math:`\frac{\left|\text{valeur approchée}-\text{valeur théorique}\right|}{\left|\text{valeur théorique}\right|}\ast 100` |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{1111}`             | 0.48437                          | 0.484961     | 0.12%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{2222}`             | --                               | 0.484782     | --                                                                                                                        |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{3333}`             | 0.73037                          | 0.730671     | 0.04%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{1122}`             | 0.12246                          | 0.122638     | 0.14%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{1133}`             | 0.18205                          | 0.182280     | 0.13%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{2233}`             | --                               | 0.182226     | --                                                                                                                        |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{1212}`             | 0.16161                          | 0.163213     | 0.98%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{2323}`             | --                               | 0.344923     | --                                                                                                                        |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{A}_{3131}`             | 0.344589                         | 0.344814     | 0.06%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{K}_{11}`               | 0.447967                         | 0.448258     | 0.06%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{K}_{22}`               | --                               | 0.448399     | --                                                                                                                        |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+
| :math:`{K}_{33}`               | 0.62115                          | 0.621319     | 0.03%                                                                                                                     |
+--------------------------------+----------------------------------+--------------+---------------------------------------------------------------------------------------------------------------------------+

On vérifie également les égalités entre les coefficients homogénéisés décrites dans le paragraphe précédent.


+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| Relation entre coefficients homogénéisés                                                                                                                                                                           | Erreur relative (%) |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`\begin{array}{c}{A}_{1111}^{\text{hom}}-{A}_{2222}^{\text{hom}}=0\end{array}`                                                                                                                               | 0.04%               |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`\begin{array}{c}{A}_{1133}^{\text{hom}}-{A}_{2233}^{\text{hom}}=0\end{array}`                                                                                                                               | 0.03%               |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`\begin{array}{c}{A}_{3131}^{\text{hom}}-{A}_{2323}^{\text{hom}}=0\end{array}`                                                                                                                               | 0.03%               |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`{A}_{3333}^{\text{hom}}-\frac{1}{|{Z}_{e}|}{\int}_{{Z}_{e}}E\phantom{\rule{0.5em}{0ex}}\mathit{dz}+2{\nu}^{2}.\left({A}_{1111}^{\text{hom}}+{A}_{1122}^{\text{hom}}\right)=0`                               | 0.34%               |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`{A}_{1133}^{\text{hom}}-\nu .\left({A}_{1111}^{\text{hom}}+{A}_{1122}^{\text{hom}}\right)=0`                                                                                                                | 1.6e-04%            |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+
| :math:`{A}_{3\alpha 3\alpha }^{\text{hom}}-\frac{2\mu }{k}{K}_{11}^{\text{hom}}=0`                                                                                                                                 | 2.7e-05%            |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+