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


Geometry
---------

We consider a one-dimensional structure (a bar whose lateral faces are subject to adiabatic conditions) of length :math:`\mathrm{2L}` occupying the domain :math:`[-L;L]`.

Since the temperature is homogeneous in the directions normal to the bar, the calculation can be considered as 1D.


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


.. csv-table::

    ":math:`\lambda \mathrm{=}2` ", "thermal conductivity"
    ":math:`\rho C\mathrm{=}2` ", "volume heat"



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

Non-linear volume source loading, temperature dependent:

:math:`s(T)\mathrm{=}2\mathrm{-}2\mathrm{\times }w\mathrm{\times }T` with :math:`w\mathrm{=}2`

The boundary conditions are adiabatic on the lateral faces and of the imposed zero temperature type at the end of the bar; a symmetry condition is implemented on the symmetry plane (which is equivalent to an adiabatic boundary condition).

The time range :math:`\mathrm{[}0.;1.\mathrm{]}` is discretized into 100 time steps (duration of each time step equal to :math:`0.01`).


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

An initial analytical state is provided. See the developments in the paragraph detailing the reference solution.