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


Geometry
---------

Structure :math:`\mathrm{2d}` is a unitary square (:math:`\mathit{LX}\mathrm{=}1m`, :math:`\mathit{LY}\mathrm{=}1m`), with a right crack leading to the right, located halfway up. (Figure). We call the left line the line in :math:`x\mathrm{=}0`, the right line the line in :math:`x\mathrm{=}\mathit{LX}`, and the bottom line the line in :math:`y\mathrm{=}0`.


.. image:: images/100000000000020E000001C38B9C795AD10E9DE2.png
    :width: 3.5433in
    :height: 3.0358in


.. _RefImage_100000000000020E000001C38B9C795AD10E9DE2.png:

**Figure** 1.1-a **: Geometry of the cracked square plate**

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:`(\mathit{LX},{\mathit{LY}}^{\text{-}}\mathrm{/}2)` (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).


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}`

Exchange coefficient between the lips of the crack :math:`h=2{\mathit{W.m}}^{\text{-2}}\mathrm{.}{K}^{\text{-1}}`


.. _RefNumPara__11653_1489988600:


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

We solve the problem on the discretized time interval :math:`\left[0\text{.}s,1\text{.}s\right]` in 5 equal time steps (of duration :math:`\Delta t=0.2s`). We take the value by default in THER_LINEAIRE of the theta-schema parameter: :math:`\theta =0.57`.

On the nodes of segment :math:`\mathit{AB}` (see Figure), the following temperature ramp is imposed:

to :math:`t=0\text{.}s`, :math:`\text{}\stackrel{̄}{T}{\text{}}^{\mathit{AB}}=10°C`; to :math:`t=1\text{.}s`, :math:`\text{}\stackrel{ˉ}{T}{\text{}}^{\mathit{AB}}\mathrm{=}20°C`

On the nodes of segment :math:`\mathit{CD}` (see Figure), the following temperature ramp is imposed:

to :math:`t=0\text{.}s`, :math:`\text{}\stackrel{̄}{T}{\text{}}^{\mathit{CD}}=20°C`; to :math:`t=1\text{.}s`, :math:`\text{}\stackrel{̄}{T}{\text{}}^{\mathit{CD}}=40°C`

Finally, on the lips of the crack, a Neumann condition, such as an exchange condition, is imposed.


.. _RefNumPara__11657_1489988600:


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

The initial state is determined by solving the stationary problem at :math:`t\mathrm{=}0\text{.}s` (with the boundary conditions given in paragraph :ref:`1.3 <RefNumPara__11653_1489988600>`)