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

Geometry
---------


.. image:: images/1000063000001A40000012B5B1AA1C85C2E5BB2D.svg
    :width: 287
    :height: 205


.. _RefImage_1000063000001A40000012B5B1AA1C85C2E5BB2D.svg:

:math:`\stackrel{ˉ}{\mathrm{AA}}\text{'}=\mathrm{2L}=2m`

:math:`x(\mathrm{M1})=0.2m`

:math:`x(\mathrm{M2})=0.8m`


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

:math:`\lambda =1W/m°C`

:math:`\rho {C}_{P}=1J/{m}^{3}°C`


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


* :math:`A:T(\mathrm{0,}t)={T}_{p}=100°C`

For :math:`t>0`


* :math:`A\text{'}:T(\mathrm{2L},t)={T}_{p}=100°C`


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

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


Details concerning the models
------------------------

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

Thermal shock requires "fine" discretization in time close to :math:`t=0`.

Since the aim of the test was to validate the various elements (different models), we chose a single discretization in time:

.. csv-table::

    "10", "not for", ":math:`[0.,1.E-3]` ", "either", ":math:`\Delta t={10}^{–4}s`"
    "9", "not for", ":math:`[1.E-\mathrm{3,}1.E-2]` ", "either", ":math:`\Delta t={10}^{–3}s`"
    "9", "not for", ":math:`[1.E-\mathrm{2,}1.E-1]` ", "either", ":math:`\Delta t={10}^{–2}s`"
    "9", "not for", ":math:`[1.E-\mathrm{1,}1.]` ", "either", ":math:`\Delta t={10}^{–1}s`"
    "10", "not for", ":math:`[1.\mathrm{,2}\mathrm{.}]` ", "either", ":math:`\Delta t={10}^{–1}s`"


Shock is defined in two different ways:


* for B modeling, this is a real shock (:math:`{T}_{p}` is discontinuous):

:math:`\{\begin{array}{}{T}_{p}^{\text{-}}(A)=0.\\ {T}_{p}^{\text{+}}(A)=100.\end{array}`


* for models :math:`A,C,D,E,F,G`, this is a linear ramp:

:math:`\{\begin{array}{}{T}_{p}{(A)}_{t=0}=0.\\ {T}_{p}{(A)}_{t={10}^{-3}}=100.\end{array}`