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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 463
    :height: 332


.. _RefImage_Object_1.svg:

The following surfaces are defined:


* Face :math:`\mathrm{YZ1}`: containing nodes :math:`\mathrm{P1},\mathrm{P3},\mathrm{P5}` and :math:`\mathrm{P7}`.


* Face :math:`\mathrm{YZ2}`: containing nodes :math:`\mathrm{P9},\mathrm{P10},\mathrm{P11}` and :math:`\mathrm{P12}`.


* Face :math:`\mathrm{XY1}`: containing nodes :math:`\mathrm{P1},\mathrm{P2},\mathrm{P9},\mathrm{P3},\mathrm{P4}` and :math:`\mathrm{P10}`.


* Face :math:`\mathrm{XY2}`: containing nodes :math:`\mathrm{P5},\mathrm{P6},\mathrm{P11},\mathrm{P7},\mathrm{P8}` and :math:`\mathrm{P12}`.


* Face :math:`\mathrm{XZ1}`: containing nodes :math:`\mathrm{P3},\mathrm{P4},\mathrm{P10},\mathrm{P7},\mathrm{P8}` and :math:`\mathrm{P12}`.


* Face :math:`\mathrm{XZ2}`: containing nodes :math:`\mathrm{P1},\mathrm{P2},\mathrm{P9},\mathrm{P5},\mathrm{P6}` and :math:`\mathrm{P11}`.

and the following items:


* Element :math:`\mathrm{M1}`: containing nodes :math:`\mathrm{P1},\mathrm{P2},\mathrm{P3},\mathrm{P4},\mathrm{P5},\mathrm{P6},\mathrm{P7}` and :math:`\mathrm{P8}`.


* Element :math:`\mathrm{M2}`: containing nodes :math:`\mathrm{P2},\mathrm{P9},\mathrm{P4},\mathrm{P10},\mathrm{P6},\mathrm{P11},\mathrm{P8}` and :math:`\mathrm{P12}`.


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

Two materials are used:


* Material :math:`\mathit{MAT1}` attributed to element :math:`\mathrm{M1}`:

Young's module: :math:`\mathit{E1}\mathrm{=}200000\mathit{MPa}`

Poisson's ratio: :math:`{\nu }_{1}\mathrm{=}0.3`


* Material :math:`\mathit{MAT2}` attributed to element :math:`\mathrm{M2}`:

Young's module: :math:`\mathrm{E2}=100000\mathrm{MPa}`

Poisson's ratio: :math:`{\nu }_{2}=0.3`


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

**First calculation:**

It's a simple traction calculation following direction :math:`X`:


* A linear elastic deformation :math:`{\varepsilon }_{\mathit{xx}}\mathrm{=}1` is imposed on the surface :math:`\mathrm{YZ2}`.


* Surface :math:`\mathrm{YZ1}` does not move in the :math:`X` direction.

**Second calculation:**

It's a simple traction calculation following direction :math:`Y`:


* A linear elastic deformation :math:`{\varepsilon }_{\mathit{yy}}\mathrm{=}1` is imposed on the surface :math:`\mathrm{XZ2}`.


* Surface :math:`\mathrm{XZ1}` does not move in the :math:`Y` direction.