Move jump
===================

The joints are intended to represent two facing faces, they only involve interpolation functions and the integration points of the corresponding *surfacic* (in 3D) or *lineic* (in 2D) elements (in 2D):

In 2D: for joint QUAD4 (or joint HYME QUAD8), the line element is SEG2

In 3D: for joint PENTA6 (or joint HYME PENTA15) the surface element is TRIA3

     for joint HEXA8 (or joint HYME HEXA20) the surface element is QUAD4.

We call :math:`{N}_{n}` the shape function of the node :math:`n` of the surface element [1] _
. :math:`{U}^{\text{+n}}` and :math:`{U}^{\text{-n}}` respectively designate the nodal movements of the segments :math:`{\Gamma }^{\text{+}}` and :math:`{\Gamma }^{\text{-}}` in 2D or the faces :math:`{S}^{\text{+}}` and :math:`{S}^{\text{-}}` in 3D.

In the local coordinate system, the displacement jump :math:`\delta` is discretized based on the form functions :math:`{N}_{n}`. At Gauss point :math:`g`, it is expressed as the difference in the displacements of the + and - faces (or segments):

:math:`{\delta }_{g}=\sum _{n=1}^{\mathrm{Nb}}R({U}^{+n}-{U}^{-n}){N}_{n}^{g}`

where :math:`\mathrm{Nb}` is the number of nodes of the surface element and where :math:`R` transition matrix in 2D, in 3D, which allows the nodal movements to be expressed in the local coordinate system. We can summarize the previous expression in a matrix :math:`{M}_{g}^{U}` which acts on the vector of the nodal displacements of the element: :math:`U`, to build the displacement jump in the local coordinate system:

 :math:`{\delta }_{g}={M}_{g}^{U}U`

The :math:`{M}_{g}^{U}` matrix is of dimension :math:`\mathrm{ndim}\times {\mathrm{Nddl}}_{U}`, with :math:`{\mathrm{Nddl}}_{U}` number of *mechanic* degrees of freedom:

:math:`{\mathrm{Nddl}}_{U}=8` for the 2D joint,

:math:`{\mathrm{Nddl}}_{U}=24` for the 3D joint HEXA

:math:`{\mathrm{Nddl}}_{U}=18` for the 3D joint PENTA

:math:`{\mathrm{Nddl}}_{U}=12` for the HYME 2D joint

:math:`{\mathrm{Nddl}}_{U}=48` for the joint HYME 3D HEXA

:math:`{\mathrm{Nddl}}_{U}=36` for the joint HYME 3D PENTA


.. _RefHeading__33316266:


.. [1] Subsequently, we use the generic term: "surface" for 2D as for 3D.