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

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

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

where :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`


+------------------------------+-----------------------------------------------------+--------+--------+--------+--------+--------+
|                              |Temperature in :math:`°C` to :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     |
+------------------------------+-----------------------------------------------------+--------+--------+--------+--------+--------+

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


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

:math:`t=4\mathrm{320s}(\mathrm{1.2hr})`: temperature at the following points:


* in :math:`x=0.6`: for :math:`y=0.6,1.5,2.4,3.0`


* in :math:`x=1.5`: for :math:`y=0.6,1.5,2.4,3.0`


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

Analytical solution.


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


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