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


Geometry
---------


.. image:: images/1000000000000256000001261087239B6AF2363C.png
    :width: 6.2217in
    :height: 3.0555in


.. _RefImage_1000000000000256000001261087239B6AF2363C.png:

:math:`r(A)=417\mathrm{mm}`

:math:`r(B)=496\mathrm{mm}`

:math:`r(\mathrm{M1})=443.43\mathrm{mm}`

:math:`r(\mathrm{M2})=469.67\mathrm{mm}`


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

:math:`\lambda =19.97W/m°C`

:math:`\rho {C}_{p}=4.89488{10}^{6}J/{m}^{3}°C`


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

Exchange :math:`\lambda \frac{\partial T}{\partial n}{\mid }_{r={r}_{i}}=h({T}_{\mathrm{ext}}-T(r,t))`

with :math:`\begin{array}{}h=\text{40000}W/{m}^{2}°C\\ {T}_{\mathrm{ext}}{(A)}_{t=\mathrm{0s}}=289°C\\ {T}_{\mathrm{ext}}{(A)}_{t=\mathrm{12s}}=20°C\end{array}`


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

:math:`T(r,0)=289°C` for everything :math:`r`

Discretization in time (:math:`t`):


.. csv-table::

    "12", "not for", ":math:`[0.,12.]` ", "either", ":math:`\Delta t=1.s`"
    "2", "not for", ":math:`[12.,20.]` ", "either", ":math:`\Delta t=4.s`"
    "4", "not for", ":math:`[20.,100.]` ", "either", ":math:`\Delta t=20.s`"
    "2", "not for", ":math:`[100.,200.]` ", "either", ":math:`\Delta t=50.s`"
    "2", "not for", ":math:`[200.,400.]` ", "either", ":math:`\Delta t=100.s`"
    "8", "not for", ":math:`[400.,2000.]` ", "either", ":math:`\Delta t=200.s`"