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

Oscillations in contact pressures may occur in certain cases, in particular for structures where the interface intersects pentahedra, under non-uniform loading.

This is due to non-compliance with the LBB condition [:ref:`bib3 <bib3>`] [:ref:`bib4 <bib4>`]. This phenomenon of oscillations is comparable to that encountered in incompressibility [:ref:`bib5 <bib5>`]. Physically, in the case of contact, this amounts to wanting to impose contact at too many points of the interface (over-stress), making the system hyperstatic. To release it, you have to restrict the space of the Lagrange multipliers, as is done in [:ref:`bib6 <bib6>`] for the Dirichlet conditions with X- FEM. Such algorithms present P0 segments, which slow down convergence. A good algorithm should minimize the occurrence of such segments. The algorithm proposed by Moës [:ref:`bib6 <bib6>`] to reduce oscillations is adapted to the 3D case (algorithm version 1). This algorithm has been improved to make it more physical and more efficient (algorithm version 2). A comparison of the two versions is made.

It should be noted that these parasitic oscillations are not reproducible in the current version of Code_Aster: one of the two algorithms is systematically chosen (2 is used by default), and even overloading the code to use none would lead to a zero pivot. We illustrate them in this documentation with results from another formulation (formulation with edges [:ref:`bib7 <bib7>`]), which has now been resolved.

For hexahedra or quadrangles cut horizontally, there are no P0 segments.


Geometry
---------

The structure is a straight parallelepiped with a square and healthy base. The dimensions of the block are: :math:`\mathrm{LX}=5m`, :math:`\mathrm{LY}=20m`, and :math:`\mathrm{LZ}=20m`. It has no cracks [:ref:`Figure 1.1-1 <Ref116447775>`].

The interface is introduced by level sets directly into the command file using the DEFI_FISS_XFEM [:ref:`U4.82.08 <U4.82.08>`] operator. The interface is present within the structure through its representation by the level sets. The normal level set (:math:`\mathrm{LSN}`) allows you to define a flat, non-leaning interface that completely crosses the elements, using the following equation:


.. _RefEquation 1.1-1:

:math:`\mathrm{LSN}=Z-17.5` eq 1.1-1


.. image:: images/10000000000001EB000001B8A355437545519CC5.png
    :width: 3.3244in
    :height: 2.9791in


.. _RefImage_10000000000001EB000001B8A355437545519CC5.png:


.. _Ref116447775:

Figure 1.1-1: Interface geometry and positioning


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

Young's module: :math:`E=100\mathit{GPa}`.

Poisson's ratio: :math:`\nu =0`.


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

The underside is embedded.

The upper face is subjected to parabolic pressure whose expression is:


.. _RefEquation 1.3-1:

:math:`\mathrm{pression}=(100-\frac{{(Y-10)}^{2}}{2})\frac{E}{{10}^{6}}\mathrm{Pa}` eq 1.3-1

Displacements along the :math:`x` and :math:`y` axes are blocked for the nodes on the upper surface.


Bibliography
-------------



.. _Ref1151529311:

1. Massin P., Ben Dhia H., Zarroug M.: Contact elements derived from a continuous hybrid formulation, *Code_Aster* Reference Manual, [:ref:`R5.03.52 <R5.03.52>`]



.. _Ref1151529531:

2. Massin P., Geniaut S.: Method X- FEM, *Code_Aster* Reference Manual, [:ref:`R7.02.12 <R7.02.12>`]



.. _Ref1164449761:

3. Babuška I.: The finite element method with lagrangian multipliers, Numerische Math 20, 179-192, 1973



.. _Ref1164449831:

4. Barbosa H., Hugues T.: Finite element method with lagrange multipliers on the boundary. Circumventing the Babuška-Brezzi condition, Comp. Meth. Applied Mech Engrg. 85 (1), 109-128, 1991



.. _Ref1167204891:

5. Chapelle D., Bathe K.J.: The INF-Sup Test, Computers & Structures 47 (4/5), 537-545, 1993


6. Moës N., Béchet E., Tourbier M.: Imposing Dirichlet boundary conditions in the extended finite element method, Int. J.Numer. Meth. Engng, 2006, Vol. 67 (12), 1641-1699.


7. Geniaut S., Massin P., Moës N., A stable 3D contact formulation using X- FEM, European Journal of Computational Mechanics, Vol.16, No. 2, No. 2, Pages 259-276, 2007.