Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

:math:`T(x,y,t)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }{B}_{n}\mathrm{cos}\frac{(\mathrm{2n}-1)\pi x}{2{L}_{x}}\mathrm{sin}\frac{j\pi y}{{L}_{y}}\mathrm{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]`

where :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`


+------------------------------+-----------------------------------------------------------------------------+--------+--------+---------+---------+---------+---------+---------+---------+--------+
|                              |**Temperature in** :math:`°F` **to** :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     |
+------------------------------+-----------------------------------------------------------------------------+--------+--------+---------+---------+---------+---------+---------+---------+--------+

Reference values are obtained with :math:`n=j=1000`


Benchmark results
----------------------

:math:`t=\mathrm{1.2hr}(\mathrm{4320s})`: temperature at the following points:


* in :math:`x=0.0`: for :math:`y=0.6,1.5,2.7`,


* in :math:`x=0.9`: for :math:`y=0.6,1.5,2.7`,


* in :math:`x=1.8`: for :math:`y=0.6,1.5,2.7`.


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


* 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, no. 3, pp 481-494, 1974.