Benchmark solution
=====================


Calculation method
-----------------

The reference solution is obtained by refining the modeling A mesh (classical axisymmetric elements with mesh crack): regular mesh composed of :math:`1000\mathrm{\times }2000` QUAD8 (instead of :math:`100\times 200` QUAD8 for mesh A)


Reference quantities and results
------------------------

The temperature is tested at the end of the last time step (:math:`t=1\text{.}s`) at points :math:`{P}^{\text{+}}(\theta )` and :math:`{P}^{\text{-}}(\theta )` (see Figure).


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**"
    "Point :math:`{P}^{\text{+}}(\theta )` - :math:`\mathit{TEMP}` ", "'AUTRE_ASTER'", ":math:`\mathrm{23,559884847913}°C`"
    "Point :math:`{P}^{\text{+}}(\theta )` - :math:`\mathit{TEMP}` ", "'AUTRE_ASTER'", ":math:`\mathrm{15,592470476233}°C`"


Since the problem is axisymmetric, the values tested cannot vary with :math:`\theta`. These values are then tested with:


* :math:`\theta \mathrm{=}0` for A and B models (:math:`\text{FEM AXIS}` and :math:`\text{X-FEM AXIS}` respectively)


* :math:`\theta \mathrm{=}\pi \mathrm{/}4` for C and D models (:math:`\text{FEM 3D}` and :math:`\text{X-FEM 3D}` respectively)