Reference problem
=====================


Geometry
---------

Structure :math:`\mathrm{2d}` is a unit square: :math:`\mathit{LX}\mathrm{=}1m` and :math:`\mathrm{LY}=1m` (Figure).


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+ .. image:: images/10000000000001D9000001C359B1BA019214653F.png                                                                                                   + .. image:: images/100000000000020E000001C38B9C795AD10E9DE2.png                                                                                                                +
|    :width: 2.3748in                                                                                                                                              |    :width: 2.6421in                                                                                                                                                           |
+    :height: 2.2642in                                                                                                                                             +    :height: 2.2642in                                                                                                                                                          +
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+ Figure 1.1-1: Healthy Plate Geometry                                                                                                                             + Figure 1.1-2: Plate geometry with fictional crack                                                                                                                             +
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For the purposes of the test case, the same domain is also considered in the presence of a fictional crack, straight and emerging on the right, located halfway up (Figure). Note :math:`{P}^{\text{+}}` the coordinate point :math:`\left(\mathit{LX},{\mathit{LY}}^{\text{+}}/2\right)` (located on the upper lip), :math:`{P}^{\text{-}}` the coordinate point :math:`\left(\mathit{LX},{\mathit{LY}}^{\text{-}}/2\right)` (located on the lower lip), and :math:`Q` the coordinate point :math:`\left(\mathit{LX}/\mathrm{2,}\mathit{LY}/2\right)` (located at the tip of the crack).

*Note:*

*This crack is qualified as fictional because it does not constitute a step in the domain's border, and therefore has no impact on the solution of the reference problem. Its presence is only due to the computer validation of the tested functionality.*


Material properties
----------------------

Thermal conductivity: :math:`\lambda =1{\mathit{W.m}}^{\text{-1}}\mathrm{.}{K}^{\text{-1}}`

Calorific volume capacity: :math:`\rho {C}_{p}=2\mathit{J.m}-{\mathrm{3.K}}^{-1}`


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Boundary conditions and loads
-------------------------------------

We impose a temperature :math:`\text{}\stackrel{̄}{T}{\text{}}^{\text{inf}}=10°C` on the nodes of segment :math:`\mathit{AB}` and :math:`\text{}\stackrel{̄}{T}{\text{}}^{\text{sup}}=20°C` on the nodes of segment :math:`\mathit{CD}` (see Figure).


.. _RefNumPara__11657_1489988600:


Initial conditions
--------------------

None (the problem is stationary)