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


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

The reference solution is analytical: the test is elementary, the cube is homogeneous deformation according to :math:`z` (zero Poisson's ratio). The force is therefore distributed according to the values of the shape functions on the nodes of the interface.


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

To move :math:`0.2\mathit{mm}` down the :math:`A` cube, we need to find (for QUAD4):


.. csv-table::

    "*Cube*", "*Point*", "DEPL
    :math:`\mathit{DZ}` "," REAC_NODA
    :math:`\mathit{DZ}`"
    ":math:`A` "," :math:`\mathit{NH1}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH2}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH3}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH4}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH9}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH10}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH11}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH12}` ", "-0,1", "10000"
    ":math:`A` "," :math:`\mathit{NH21}` ", "-0,1", "10000"
    ":math:`B` "," :math:`\mathit{NB5}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB6}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB7}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB8}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB17}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB18}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB19}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB20}` ", "-0,1", "- 10000"
    ":math:`B` "," :math:`\mathit{NB26}` ", "-0,1", "- 10000"


To move :math:`0.2\mathit{mm}` down cube A, we need to find (for QUAD8):


.. csv-table::

    "*Cube*", "*Point*", "DEPL
    :math:`\mathit{DZ}` "," REAC_NODA
    :math:`\mathit{DZ}`"
    ":math:`A` "," :math:`\mathit{NH1}` ", "-0,1", "- 10000/ 3"
    ":math:`A` "," :math:`\mathit{NH2}` ", "-0,1", "- 10000/ 3"
    ":math:`A` "," :math:`\mathit{NH3}` ", "-0,1", "- 10000/ 3"
    ":math:`A` "," :math:`\mathit{NH4}` ", "-0,1", "- 10000/ 3"
    ":math:`A` "," :math:`\mathit{NH9}` ", "-0,1", ", "(4* 10000) /3"
    ":math:`A` "," :math:`\mathit{NH10}` ", "-0,1", ", "(4* 10000) /3"
    ":math:`A` "," :math:`\mathit{NH11}` ", "-0,1", ", "(4* 10000) /3"
    ":math:`A` "," :math:`\mathit{NH12}` ", "-0,1", ", "(4* 10000) /3"
    ":math:`B` "," :math:`\mathit{NB5}` ", "-0,1", "10000/ 3"
    ":math:`B` "," :math:`\mathit{NB6}` ", "-0,1", "10000/ 3"
    ":math:`B` "," :math:`\mathit{NB7}` ", "-0,1", "10000/ 3"
    ":math:`B` "," :math:`\mathit{NB8}` ", "-0,1", "10000/ 3"
    ":math:`B` "," :math:`\mathit{NB17}` ", "-0,1", ", "- (4* 10000) /3"
    ":math:`B` "," :math:`\mathit{NB18}` ", "-0,1", ", "- (4* 10000) /3"
    ":math:`B` "," :math:`\mathit{NB19}` ", "-0,1", ", "- (4* 10000) /3"
    ":math:`B` "," :math:`\mathit{NB20}` ", "-0,1", ", "- (4* 10000) /3"


To move :math:`0.2\mathit{mm}` down the :math:`A` cube, we need to find (for QUAD9):


.. csv-table::

    "*Cube*", "*Point*", "DEPL
    :math:`\mathit{DZ}` "," REAC_NODA
    :math:`\mathit{DZ}`"
    ":math:`A` "," :math:`\mathit{NH1}` ", "-0,1", "10000/ 9"
    ":math:`A` "," :math:`\mathit{NH2}` ", "-0,1", "10000/ 9"
    ":math:`A` "," :math:`\mathit{NH3}` ", "-0,1", "10000/ 9"
    ":math:`A` "," :math:`\mathit{NH4}` ", "-0,1", "10000/ 9"
    ":math:`A` "," :math:`\mathit{NH9}` ", "-0,1", ", "(4* 10000)/9"
    ":math:`A` "," :math:`\mathit{NH10}` ", "-0,1", ", "(4* 10000)/9"
    ":math:`A` "," :math:`\mathit{NH11}` ", "-0,1", ", "(4* 10000)/9"
    ":math:`A` "," :math:`\mathit{NH12}` ", "-0,1", ", "(4* 10000)/9"
    ":math:`A` "," :math:`\mathit{NH21}` ", "-0,1", ", "(16 * 10000)/9"
    ":math:`B` "," :math:`\mathit{NB5}` ", "-0,1", "- 10000/ 9"
    ":math:`B` "," :math:`\mathit{NB6}` ", "-0,1", "- 10000/ 9"
    ":math:`B` "," :math:`\mathit{NB7}` ", "-0,1", "- 10000/ 9"
    ":math:`B` "," :math:`\mathit{NB8}` ", "-0,1", "- 10000/ 9"
    ":math:`B` "," :math:`\mathit{NB17}` ", "-0,1", ", "- (4* 10000)/9"
    ":math:`B` "," :math:`\mathit{NB18}` ", "-0,1", ", "- (4* 10000)/9"
    ":math:`B` "," :math:`\mathit{NB19}` ", "-0,1", ", "- (4* 10000)/9"
    ":math:`B` "," :math:`\mathit{NB20}` ", "-0,1", ", "- (4* 10000)/9"
    ":math:`B` "," :math:`\mathit{NB26}` ", "-0,1", ", "- (16 * 10000)/9"


For the continuous formulation, contact pressures LAGS_C are tested in addition to the REAC_NODA nodal reactions. These are the true pressure values. So we have to find a pressure of :math:`p\mathrm{=}\mathit{E.}(0.1\mathrm{/}2)\mathrm{=}10000` on the nodes :math:`\mathit{NH1}`, :math:`\mathit{NH2}`, :math:`\mathit{NH3}`,, :math:`\mathit{NH4}`, :math:`\mathit{NH9}`, :math:`\mathit{NH10}`, :math:`\mathit{NH11}`, :math:`\mathit{NH12}` and :math:`\mathit{NH21}`.


Uncertainties about the solution
----------------------------

None (analytical solution).