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

Calculation method used for the reference solution
--------------------------------------------------------

In each case, a preliminary calculation is carried out with the keyword LIAISON_DDL to introduce linear relationships between degrees of freedom. This calculation is used as a reference to the calculation with the LIAISON_MAIL keyword that generates these linear relationships.

To get the desired linear relationships with LIAISON_MAIL, we write:

**Case 1:**

.. code-block:: text

    LIAISON_MAIL :( NOEUD_2: E MAILLE_1 :Q1
CENTRE: B ANGL_NAUT: 90. TRAN :( -5. 0.))

This means that we eliminate the 2 degrees of freedom of the node :math:`E` according to the degrees of freedom of the point :math:`E\text{'}` obtained when we make :math:`E` undergo a rotation of 90 degrees around :math:`B` and then a vector translation (—5.0). So :math:`E\text{'}` is in the middle of :math:`\mathit{CD}`. The displacement vector of :math:`E` is identified (after rotation of 90 degrees) with that of :math:`E\text{'}`. So we get the 2 equations:

.. code-block:: text

    DX (E) = DY (E') = 0.5 DY (C) + 0.5 DY (D)

    DY (E) = - XX (E') = -0.5 XX (C) - 0.5 XX (D)

**Case 2:**

.. code-block:: text

    LIAISON_MAIL :( MAILLE_2: S1 MAILLE_1 :( Q1, Q2)
DDL_2: 'DNOR' DDL_1: 'DNOR'
CENTRE: B ANGL_NAUT: 180. TRAN :( +5. +10.))

This means that we eliminate the normal displacement of the nodes :math:`B` and :math:`E` (nodes of the segment :math:`\mathit{S1}`) according to the degrees of freedom of the points :math:`B\text{'}` and :math:`E\text{'}` obtained when we make :math:`B` and :math:`E` undergo a rotation of 180 degrees around :math:`B` and then a vector translation :math:`(+\mathrm{5,}+10)`. So :math:`B\text{'}` is in the middle of :math:`\mathit{CF}` and :math:`E\text{'}` is in the middle of :math:`\mathit{DC}`. The normal movement of B is identified (after rotating 180 degrees) to that of :math:`B\text{'}`. We're doing the same for :math:`B\text{'}`. We then obtain the 2 equations:


.. code-block:: text

    :math:`\mathit{DY}(E)\mathrm{=}\mathrm{-}\mathit{DY}(E\text{'})\mathrm{=}\mathrm{-}0.5\mathit{DY}(C)\mathrm{-}0.5\mathit{DY}(D)`

    :math:`\mathit{DY}(B)\mathrm{=}\mathrm{-}\mathit{DY}(B\text{'})\mathrm{=}\mathrm{-}0.5\mathit{DY}(C)\mathrm{-}0.5\mathit{DY}(F)`


Benchmark results
----------------------

We observe the :math:`\mathit{DY}` movement of the point :math:`F`:

case 1: :math:`\mathit{DY}(F)\mathrm{=}\mathrm{1.4153582447720D}+00`

case 2: :math:`\mathit{DY}(F)\mathrm{=}\mathrm{1.0561898652983D}+00`

These displacements are obtained with linear relationships between degrees of freedom introduced by the keyword LIAISON_DDL.


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

No uncertainty.