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

The reference problem is inspired by an article by F. Auricchio*et al*. :ref:`[1] <[1]>` who offers an analytical solution.


Geometry
---------

We consider a square with dimension :math:`2m\times 2m`.

The geometry can be visualized in the figure, with the mesh that will be used for both models A and B.


.. image:: images/10000000000005DB000003214C69DF1A38759B8B.svg
    :width: 430
    :height: 229


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Figure 1.1-1: Geometry and meshing


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

The material is elastic and almost incompressible, that is to say that its Poisson's ratio tends to 0.5:


* Modulus of elasticity: :math:`E=120\mathit{Pa}`


* Poisson's ratio: :math:`\mathrm{\nu }=0.499999`

These values may seem surprising but have no physical meaning since the test is purely mathematical.


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

The four edges of the square (:math:`\mathit{DX}=\mathit{DY}=0`) are embedded, and a volume force is applied to the entire square. This force is variable in space and has the following components:

 :math:`\mathit{FX}=\mathrm{\mu }y(\frac{-3}{2}{x}^{4}+6{x}^{2}-3{x}^{2}{y}^{2}+{y}^{2}-\frac{5}{2})-15{x}^{2}(y-1)`

 :math:`\mathit{FY}=\mathrm{\mu }x(\frac{3}{2}{y}^{4}-6{y}^{2}+3{y}^{2}{x}^{2}-{x}^{2}+\frac{5}{2})-3{y}^{2}-5{x}^{3}`


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