D modeling
==============


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

The symmetry of the problem makes it possible to represent only a quarter of the model in 3D: The sphere and the block are represented respectively by a quarter sphere and a quarter of a cylinder, meshed with 3D solid elements CUB8.

A node-mesh contact is defined between the sphere and the block.

An imposed displacement load is applied to the upper part of the sphere, which is rigidified by a high Young's modulus.


.. image:: images/100000000000015E00000190A4972D35F145E88B.png
    :width: 3.6453in
    :height: 4.1661in


.. _RefImage_100000000000015E00000190A4972D35F145E88B.png:

**Boundary condition:**


.. csv-table::

    "* Symmetry conditions:", "the nodes located in plane :math:`(O,y,z)` (node groups :math:`«\mathrm{SBYZ}»` and :math:`«\mathrm{SSYZ}»`) are blocked in the direction :math:`X` (DX= 0),"
    "", "the nodes located in plane :math:`(O,x,y)` (node groups :math:`«\mathrm{SBXY}»` and :math:`«\mathrm{SSXY}»`) are blocked in the direction :math:`Z` (DZ= 0)."
    "* Embedding the base:", "the nodes of :math:`«\mathrm{BASE}»` (plane :math:`Y=0.`) are blocked in the directions :math:`X`, :math:`Y`, and :math:`Z` (DX = DY = DZ = 0)."



* Rigid body movements are suppressed by imposing a following link y between node :math:`E` belonging to the sphere and node :math:`S` belonging to the massif.

**Loads:**

An imposed displacement is applied to the upper part of the sphere (group of nodes :math:`«\mathrm{CHIMPO}»`) in the direction :math:`Y`: Loading from 0 to :math:`–100.\mathrm{mm}`


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

Number of knots: 6993

Number of meshes and type: 5544 HEXA8, 407 PENTA6 and 191 QUAD4


Tested values
---------------


.. csv-table::

    "**Identification**", "**Movations**", "**Reference**", "**Aster**", "**% tolerance**"
    "Reaction :math:`(N)` "," :math:`d=–20\mathrm{mm}` ", "—3.201351E+06", "-3.82828724E+06", "25"
    "Reaction :math:`(N)` "," :math:`d=–40\mathrm{mm}` ", "—6.402702E+06", "-7.38942843E+06", "20"
    "Reaction :math:`(N)` "," :math:`d=–60\mathrm{mm}` ", "—9.604053E+06", "-1.06420713E+07", "15"
    "Reaction :math:`(N)` "," :math:`d=–80\mathrm{mm}` ", "—1.280540E+07", "-1.28912992E+07", "10"
    "Reaction :math:`(N)` "," :math:`d=–100\mathrm{mm}` ", "—1.600675E+07", "-1.56376456E+07", "5"



notes
---------

The results are almost identical to those of modeling C. But the calculation is even more tedious because a quarter of the sphere is meshed.