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


.. _RefNumPara__707_852783072:


Geometry
---------

The structure, represented in the Figure, is a cylinder of radius :math:`R=1m` and height :math:`H=2m` including a plane crack on the section located halfway up. (included in equation plan :math:`y=H/2`). The crack is open, and has as geometric support the crown with inner radius :math:`R/2` and outer radius :math:`R` (crown filled in red in Figure).

We call the "lower disk" the disk included in equation plane :math:`y\mathrm{=}0`, and "upper disk" the disk included in equation plane :math:`y=H`.


.. image:: images/10000000000003AB000001F330D03579359A18DD.png
    :width: 5.9059in
    :height: 3.1343in


.. _RefImage_10000000000003AB000001F330D03579359A18DD.png:

Figure 1.1-1: Problem geometry

Finally, we note :math:`{P}^{\text{+}}(\theta )` the coordinate point :math:`\left(R\mathrm{cos}\theta ,{H}^{\text{+}}/2,R\mathrm{sin}\theta \right)` (located on the upper lip), and :math:`{P}^{\text{-}}(\theta )` the coordinate point :math:`\left(R\mathrm{cos}\theta ,{H}^{\text{-}}/2,R\mathrm{sin}\theta \right)` (located on the lower lip)


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

Thermal conductivity: :math:`\lambda \mathrm{=}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}`


.. _RefNumPara__11653_1489988600:


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

We solve the problem on the discretized time interval :math:`\left[0.s,1.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 the lower disk (*cf.* paragraph :ref:`1.1 <RefNumPara__707_852783072>`) 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{̄}{T}{\text{}}^{\text{sup}}=20°C`

On the nodes of the upper disk (*cf.* paragraph :ref:`1.1 <RefNumPara__707_852783072>`) the following temperature ramp is imposed:

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


.. _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>`)