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


Geometry
---------


.. image:: images/10000D56000030930000174132DFB8A3C457790A.svg
    :width: 626
    :height: 300


.. _RefImage_10000D56000030930000174132DFB8A3C457790A.svg:


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

:math:`E=0`

:math:`\nu =0`

The finite elements present in this problem are only used to define the degrees of freedom carried by the nodes. Their rigidity must be zero.


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

In this problem, we define "solid" node groups:


* in 2D:


* :math:`A,B,C,D`


* :math:`\mathrm{E1},\mathrm{E2},\mathrm{E3}`


* in 3D:


* :math:`F,G,H,I,J,K,L,M`


* :math:`O,N,P`


* :math:`\mathrm{Q1},\mathrm{Q2},\mathrm{Q3}`

For each of these groups of nodes, partial displacements are imposed so that the "solids" move while respecting:


.. csv-table::

    "In 2D:", ":math:`\{\begin{array}{ccc}\mathrm{translation}& :& T(A)=T(\mathrm{E1})=(\begin{array}{}2.\\ 3.\end{array})\\ \mathrm{rotation}& :& \theta (A)=\theta (\mathrm{E1})=0.01\end{array}`"
    "In 3D:", ":math:`\{\begin{array}{ccc}\mathrm{translation}& :& T(F)=T(N)=T(\mathrm{Q1})=(\begin{array}{}2.\\ 3.\\ 4.\end{array})\\ \mathrm{rotation}& :& \theta (F)=\theta (N)=\theta (\mathrm{Q1})=(\begin{array}{}0.001\\ 0.002\\ 0.003\end{array})\end{array}`"


The imposed trips chosen to lead to the "solid" trips sought are:


.. csv-table::

    "2D", ":math:`\mathrm{DX}(A)=2.` "," :math:`\mathrm{DX}(\mathrm{E1})=2.`"
    "", ":math:`\mathrm{DY}(A)=3.` "," :math:`\mathrm{DY}(\mathrm{E1})=3.`"
    "", ":math:`\mathrm{DY}(B)=3.001` ", "(+ :math:`\mathrm{DRZ}(\mathrm{E1})=0.001` for B modeling)"
    "", "", ""
    "3D", ":math:`\mathrm{DX}(F,N,\mathrm{Q1})=2.` ", ""
    "", ":math:`\mathrm{DY}(F,N,\mathrm{Q1})=3.` ", ""
    "", ":math:`\mathrm{DZ}(F,N,\mathrm{Q1})=4.` ", ""
    "", "", ""
    "", ":math:`\mathrm{DY}(J,O)=2.002` ", ""
    "", ":math:`\mathrm{DY}(J,O)=2.999` ", ""
    "", "", ""
    "", ":math:`\mathrm{DX}(I)=1.997` ", ""
    "", "", ""
    "+", ":math:`\mathrm{DRZ}(N)=0.003` ", "for B modeling"
    "", ":math:`\mathrm{DRX}(\mathrm{Q1})=0.001` ", ""
    "", ":math:`\mathrm{DRY}(\mathrm{Q1})=0.002` ", ""
    "", ":math:`\mathrm{DRZ}(\mathrm{Q1})=0.003` ", ""