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}{B}_{n}\cos\frac{(\mathrm{2n}-1)\pi x}{2{L}_{x}}\sin\frac{j\pi y}{{L}_{y}}\exp\left[-(\frac{{\lambda}_{x}{(\mathrm{2n}-1)}^{2}{\pi}^{2}}{4{L}_{x}^{2}}+\frac{{\lambda}_{y}{j}^{2}{\pi}^{2}}{{L}_{y}^{2}})t\right]`

où :math:`{B}_{n}=\left[\frac{8{T}_{i}}{{\pi}^{2}j(\mathrm{2n}-1)}{(-1)}^{n+2}[{(-1)}^{j}-1]-32\right]\frac{5}{9}`         :math:`{T}_{i}=\frac{5}{9}{T}_{0}+32`


+-------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                   | **Température en** :math:`°F` **à** :math:`t=1.2\mathrm{hr}(\mathrm{4320s})`                                                                                                    |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 2.7               | –15,6151                                                                     | –15,6480 | –15,7455 | –15,9049 | –16,1211 | –16,3876 | –16,6964 | –17,0381 | –17,4022 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 2.4               | –15,6462                                                                     | –15,6786 | –15,7748 | –15,9318 | –16,1449 | –16,4076 | –16,7120 | –17,0487 | –17,4076 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 2.1               | –15,7391                                                                     | –15,7700 | –15,8620 | –16,0122 | –16,2160 | –16,4673 | –16,7584 | –17,0805 | –17,4238 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 1.8               | –15,8921                                                                     | –15,9208 | –16,0058 | –16,1447 | –16,3333 | –16,5657 | –16,8349 | –17,1328 | –17,4503 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 1.5               | –16,1025                                                                     | –16,1279 | –16,2035 | –16,3269 | –16,4944 | –16,7009 | –16,9401 | –17,2048 | –17,4869 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 1.2               | –16,3655                                                                     | –16,3869 | –16,4506 | –16,5547 | –16,6959 | –16,8700 | –17,0716 | –17,2947 | –17,5325 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 0.9               | –16,6744                                                                     | –16,6911 | –16,7409 | –16,8222 | –16,9325 | –17,0685 | –17,2261 | –17,4004 | –17,5862 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 0.6               | –17,0203                                                                     | –17,0318 | –17,0660 | –17,1218 | –17,1975 | –17,2909 | –17,3991 | –17,5187 | –17,6462 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 0.3               | –17,3923                                                                     | –17,3982 | –17,4156 | –17,4440 | –17,4825 | –17,5300 | –17,5851 | –17,6459 | –17,7108 | –17,7778 |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+
| 0.0               | –17,7778                                                                     | –17,7778 | –17,7778 | –17,7778 | –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      | 1.8      | 2.1      | 2.4      | 2.7      |
+-------------------+------------------------------------------------------------------------------+----------+----------+----------+----------+----------+----------+----------+----------+----------+

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


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

:math:`t=\mathrm{1.2hr}(\mathrm{4320s})` : température aux points suivants :


*   en :math:`x=0.0` : pour :math:`y=0.6,1.5,2.7` ,


*   en :math:`x=0.9` : pour :math:`y=0.6,1.5,2.7` ,


*   en :math:`x=1.8` : pour :math:`y=0.6,1.5,2.7` .


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

Solution analytique.


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


*   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.