D modeling — Multiple 2D interfaces
===========================================

The purpose of this modeling is to test the functioning of multi-crack X-FEM for a :math:`\mathrm{2D}` structure.


Characteristics of the mesh
----------------------------

We choose to model structure :math:`\mathrm{2D}` by a mesh composed of :math:`2\mathrm{\times }10` elements QUAD4, as shown on [].


.. image:: images/10000000000000D00000024862AF719EDD597963.png
    :width: 1.061in
    :height: 2.9862in


.. _RefImage_10000000000000D00000024862AF719EDD597963.png:

Figure 5.1-1: Meshing for modeling D


Boundary conditions
----------------------

The boundary conditions are similar to those imposed for modeling A, obviously adapted for case :math:`\mathrm{2D}`. The nodes belonging to the median line are embedded and displacements are imposed on the nodes belonging to the extreme lines (lower and upper). So we have:

GROUP_NO = LIGMED: DX = 0 and DY = 0

GROUP_NO = LIGINF: DX = 0 and DY = - DEPY

GROUP_NO = LIGSUP: DX = 0 and DY = DEPY

We consider :math:`\mathit{DEPY}\mathrm{=}2.E\mathrm{-}3`.


Analytical resolution
---------------------

The solution of such a problem is of course still obvious: all the movements following :math:`x` are zero, all the movements following :math:`y` below the lower level-set are equal to the displacement imposed :math:`{u}_{y}` at the base of the structure, all the movements following :math:`y` between the two level sets are zero and all the movements following :math:`y` above the upper level-set are equal to imposed displacement :math:`{u}_{y}` at the top of the structure.


Tested sizes and results
------------------------------

The displacement values are tested after convergence of the iterations of the operator STAT_NON_LINE. Each time, only one node on each floor is tested and the following table is obtained where the position of each tested node is indicated on the [].


.. image:: images/10000000000000E300000294D11BCC513B93DCCD.png
    :width: 1.1307in
    :height: 3.3055in


.. _RefImage_10000000000000E300000294D11BCC513B93DCCD.png:

Figure 5.4-1: **Final mesh and positioning of the tested nodes**


.. csv-table::

    "**Identification**", "**Reference**"
    "DXpour all nodes just below the lower interface", "0.00"
    "DXpour all nodes just above the lower interface", "0.00"
    "DYpour all nodes just below the lower interface", "-2.00E-3"
    "DYpour all nodes just above the lower interface", "0.00"
    "DXpour all nodes just below the top interface", "0.00"
    "DXpour all nodes just above the top interface", "0.00"
    "DYpour all nodes just below the top interface", "0.00"
    "DYpour all nodes just above the top interface", "2.00E-3"