Modeling A
==============


Characteristics of modeling
-----------------------------------

This is an X- FEM modeling, in plane deformations, where the interfaces are defined by level functions (normal level sets noted :math:`\text{LN}`).

The equations of the level functions for the three horizontal cracks are as follows:


.. _RefEquation 3.1-1:

:math:`\text{LN}1=Y-0.5` eq 3.1-1


.. _RefEquation 3.1-2:

:math:`\text{LN}1\mathrm{=}Y\mathrm{-}1.5` eq 3.1-2


.. _RefEquation 3.1-3:

:math:`\text{LN}2=Y-2.5` eq 3.1-3


.. _RefEquation 3.1-4:

:math:`\text{LN}3=Y-3.5` eq 3.1-4


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


.. image:: images/Object_11.png
    :width: 2.6882in
    :height: 3.3445in


.. _RefSchema_Object_11.png:

**Figure 3.2-a: The modeling mesh A.**

The mesh comprises 4 cells of the QUAD4 type, represented in FIG. 3.2-a.

For example, we notice in this figure that node :math:`\mathrm{N1}` sees 2 cracks. It must therefore be enriched twice and it then has the kinematic degrees of freedom DX, DY, H1X, H1Y, H2X and H2Y. Since the contact is active, node :math:`\mathrm{N1}` also has Lagrange degrees LAGS_C and LAG2_C.

On the other hand, we notice for example that mesh :math:`\text{M1}` "sees" 4 cracks. The element associated with this mesh will therefore store the fields of the four cracks, regardless of the degrees of freedom associated with its nodes.


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

The movements at the level of the crack lips are tested. The DX displacement should follow function :math:`\text{Depl\_X}` in equation 2.1-1. The displacement DY must follow function :math:`\text{Depl\_Y}` in equation 2.1-2. The stair deformation of FIG. 3.4-a is obtained.


.. csv-table::

    "**Identification**", "**Reference**"
    "SOMM_ABSpour DX- :math:`\text{Depl\_X}` (master side)", "0"
    "SOMM_ABSpour DY- :math:`\text{Depl\_Y}` (master side)", "0"
    "SOMM_ABSpour DX- :math:`\text{Depl\_X}` (slave side)", "0"
    "SOMM_ABS for DY- :math:`\text{Depl\_Y}` (slave side)", "0"


Table 3.3-1


.. image:: images/10000000000001A500000261F61AD6896559AD99.jpg
    :width: 2.7563in
    :height: 3.9839in


.. _RefImage_10000000000001A500000261F61AD6896559AD99.jpg:


.. image:: images/10000000000001A500000261AE4D0936369D3845.jpg
    :width: 2.7563in
    :height: 3.9882in


.. _RefImage_10000000000001A500000261AE4D0936369D3845.jpg:

**Figure 3.4-a: Deformed structure.**

We test the value of :math:`{E}^{e}` produced by the POST_ERREUR operator.


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "Ee", "'ANALYTIQUE'", "2.6 107"," 0.1%"


We test the value of :math:`{\parallel u\parallel }_{{L}^{2}}` produced by the POST_ERREUR operator.


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "L2 standard, "'ANALYTIQUE'", "1.00664459137"," 0.1%"