E modeling
==============


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

It is a 3D model. The master and slave surfaces are facing each other with compatible meshes. To avoid having calculations that are too slow, we will only take a quarter of the geometry.

Young's modulus :math:`{E}_{1}\mathrm{=}{E}_{2}` and Poisson's coefficients :math:`{\nu }_{1}\mathrm{=}{\nu }_{2}` are :math:`1.0E9\mathit{Pa}` and :math:`0.2` respectively. The pressure applied to the edge of the outer ring is :math:`1.0E7+\mathrm{10E5.cos}(2\theta )(\mathit{Pa})`, :math:`\theta` being the polar angle.

The pressure does not vary in the longitudinal direction. To have a model in accordance with the plane deformations, zero longitudinal displacement will be imposed on the two upper surfaces (in red in FIG. 7.2-1). Solution displacements will be applied as boundary conditions on the other surfaces (in blue in figure 7.2-1).

The outer ring defines the master surface.


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

The mesh (Figure 7.2-1) includes:


* 448 QUAD4 meshes;


* 384 HEXA8 meshes.


.. image:: images/100000000000035100000293598FCE7FF9E9D556.png
    :width: 2.9925in
    :height: 2.322in


.. _RefImage_100000000000035100000293598FCE7FF9E9D556.png:

**Figure** 7.2-a **: The E modeling mesh**

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

Note :math:`\lambda` the contact pressure calculated analytically, :math:`({U}_{x},{U}_{y},{U}_{z})` the displacements calculated analytically.

In the case where the displacements are zero, an absolute limit equal to 2% is defined, the value of the maximum displacement Vmax=5.5E-03.

The minimum value and the maximum value of the contact pressure (LAGS_C) and the displacement components (DX, DY, DZ) are calculated for the following groups of nodes:


* GROUPE 1: N47 (5.54327719507606E-01, 2.29610059417040E-01, 0.00000000000000E+00),

N103 (5.54327719507606E-01, 2.29610059417040E-01, 4.00000000000000E-01),

N412 (5.54327719507606E-0, 2.29610059417040E-01, 1.9999999999999678E-01)


* GROUPE 2: N50 (4.24264068712436E-01, 4.24264068711421E-01, 0.00000000000000E+00),

N106 (4.24264068712436E-01, 4.24264068711421E-01, 4.00000000000000E-01),

N403 (4.24264068712436E-01, 4.24264068711421E-01, 1.99999999999678E-01)


* GROUPE 3: N53 (2.29610059417922E-01, 5.54327719507241E-01, 0.00000000000000E+00),

N109 (2.29610059417922E-01, 5.54327719507241E-01, 4.00000000000000E-01),

N394 (2.29610059417922E-01, 5.54327719507241E-01, 1.9999999999999678E-01)


.. csv-table::

    "**Identification**", "**Reference**", "**Aster**", "**tolerance**"
    "LAGS_C min from group 1", ":math:`\lambda (x,y,z)` ", "-9355226.31353", "2.10-2"
    "DX min from group 1", ":math:`{U}_{x}(x,y,z)` ", "-0.0050526512252", "2.10-2"
    "DY min from group 1", ":math:`{U}_{y}(x,y,z)` ", "-0.0020785420526", "2.10-2"
    "DZ min from group 1", ":math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"
    "LAGS_C max from group 1", ":math:`\lambda (x,y,z)` ", "-9355226.31353", "2.10-2"
    "Group 1 max DX," :math:`{U}_{x}(x,y,z)` ", "-0.0050526512252", "2.10-2"
    "DY max of group 1"," :math:`{U}_{y}(x,y,z)` ", "-0.0020785420526", "2.10-2"
    "DZ max of group 1"," :math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"
    "LAGS_C min of group 2"," :math:`\lambda (x,y,z)` ", "-9259259.25926", "2.10-2"
    "DX min from group 2"," :math:`{U}_{x}(x,y,z)` ", "-0.0037844796198", "2.10-2"
    "DY min from group 2"," :math:`{U}_{y}(x,y,z)` ", "-0.0037579927128", "2.10-2"
    "DZ min from group 2"," :math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"
    "LAGS_C max from group 2"," :math:`\lambda (x,y,z)` ", "-9259259.25926", "2.10-2"
    "Group 2 max DX," :math:`{U}_{x}(x,y,z)` ", "-0.0037844796198", "2.10-2"
    "DY max of group 2"," :math:`{U}_{y}(x,y,z)` ", "-0.0037579927128", "2.10-2"
    "Group 2 max DZ," :math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"
    "LAGS_C min from group 3", ":math:`\lambda (x,y,z)` ", "-9163292.20499", "2.10-2"
    "DX min from group 3"," :math:`{U}_{x}(x,y,z)` ", "-0.0020034145592", "2.10-2"
    "DY min from group 3"," :math:`{U}_{y}(x,y,z)` ", "-0.0048020637882", "2.10-2"
    "DZ min from group 3"," :math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"
    "LAGS_C max from group 3", ":math:`\lambda (x,y,z)` ", "-9163292.20499", "2.10-2"
    "Group 3 Max DX," :math:`{U}_{x}(x,y,z)` ", "-0.0020034145592", "2.10-2"
    "DY max of group 3"," :math:`{U}_{y}(x,y,z)` ", "-0.0048020637882", "2.10-2"
    "Group 3 max DZ," :math:`{U}_{z}(x,y,,z)` ", "0.0", "1.E-8"


**Table** 7.3-1