4. Fluid pressure gradient

The joint elements HYME have, in addition to mechanical degrees of freedom \(U\), nodal degrees of freedom of fluid pressure (one per node) noted \(P\).

For the QUAD8, the nodes following nodes 6,7,8 carry these degrees of pressure freedom. The quadratic reference element used to approximate the pressures is SEG3.

For HEXA20, the nodes 13,14,15,16,16,17,18,19,20 carry these degrees of pressure freedom. The reference surface element used to approximate the pressures is QUAD8.

For PENTA15, the nodes following 10,11,12,13,14,15 carry these degrees of pressure freedom. The reference surface element used to approximate the pressures is TRIA6.

The fluid flow law (cubic law, see [R7.01.25]) involves the pressure gradient in the direction of flow estimated at the gauss point g in a classical way:

\(\nabla {p}_{g}=\sum _{n=1}^{\mathrm{Nb}}{P}_{n}\nabla {N}_{n}^{g}\)

where \(\mathrm{Nb}\) is the number of pressure knots and \({N}_{n}^{g}\) is the value of the node shape function \(n\) at the gauss point \(g\). To simplify the writing we note:

\(\nabla {p}_{g}={M}_{g}^{P}P\)

The \({M}_{g}^{U}\) matrix is of dimension \((\mathrm{ndim}-1)\times {\mathrm{Nddl}}_{P}\): with \({\mathrm{Nddl}}_{P}\) number of degrees of freedom fluid:

\({\mathit{Nddl}}_{P}=3\) for the HYME 2D joint

\({\mathit{Nddl}}_{P}=8\) for the joint HYME 3D HEXA

\({\mathit{Nddl}}_{P}=6\) for the joint HYME 3D PENTA