Solution de référence
=====================

Méthode de calcul utilisée pour la solution de référence
--------------------------------------------------------

:math:`T(x,y,t)=\sum_{n=1}^{\infty}\sum_{j=1}^{\infty}{A}_{n}\sin\frac{n\pi x}{{L}_{x}}\sin\frac{j\pi y}{{L}_{y}}\exp\left[-(\frac{{\lambda}_{x}{n}^{2}{\pi}^{2}}{{L}_{x}^{2}}+\frac{{\lambda}_{y}{j}^{2}{\pi}^{2}}{{L}_{y}^{2}})t/\rho c\right]`

où :math:`{A}_{n}=\left[\frac{4({T}_{i})}{{\pi}^{2}jn}[{(-1)}^{n}-1][{(-1)}^{j}-1]-32\right]\frac{5}{9}` :math:`{T}_{i}=\frac{5}{9}{T}_{0}+32`


+-------------------+-------------------------------------------------------------------------------------------------------------+
|                   | Température en :math:`°C` à :math:`t=\mathrm{4320s}`                                                        |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 3.0               | -17.7778                                             | -17.5742 | -17.3905 | -17.2448 | -17.1515 | -17.1189 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 2.7               | -17.7778                                             | -17.5764 | -17.3948 | -17.2507 | -17.1581 | -17.1262 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 2.4               | -17.7778                                             | -17.5832 | -17.4077 | -17.2684 | -17.1790 | -17.1482 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 2.1               | -17.7778                                             | -17.5945 | -17.4291 | -17.2979 | -17.2137 | -17.1847 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 1.8               | -17.7778                                             | -17.6102 | -17.4590 | -17.3391 | -17.2620 | -17.2355 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 1.5               | -17.7778                                             | -17.6302 | -17.4970 | -17.3914 | -17.3235 | -17.3002 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 1.2               | -17.7778                                             | -17.6542 | -17.5426 | -17.4541 | -17.3973 | -17.3777 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 0.9               | -17.7778                                             | -17.6816 | -17.5949 | -17.5261 | -17.4819 | -17.4667 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 0.6               | -17.7778                                             | -17.7120 | -17.6526 | -17.6056 | -17.5753 | -17.5649 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 0.3               | -17.7778                                             | -17.7444 | -17.7142 | -17.6903 | -17.6749 | -17.6696 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| 0.0               | -17.7778                                             | -17.7778 | -17.7778 | -17.7778 | -17.7778 | -17.7778 |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+
| :math:`Y\uparrow` |                                                      |          |          |          |          |          |
| :math:`X\to`      | 0.0                                                  | 0.3      | 0.6      | 0.9      | 1.2      | 1.5      |
+-------------------+------------------------------------------------------+----------+----------+----------+----------+----------+

Les valeurs de référence sont obtenues avec :math:`n=j=1000`


Résultats de référence
----------------------

:math:`t=4\mathrm{320s}(\mathrm{1.2hr})` : température aux points suivants :


*   en :math:`x=0.6` : pour :math:`y=0.6,1.5,2.4,3.0`


*   en :math:`x=1.5` : pour :math:`y=0.6,1.5,2.4,3.0`


Incertitude sur la solution
---------------------------

Solution analytique.


Références bibliographiques
---------------------------


1.  J.C. Bruch Jr., G. Zyroloski, ‘Transient two-dimensional heat conduction problems solved by the finite element method’, Int. J. num. Meth. Engng, vol 8, n°3, pp 481-494, 1974.