7. F modeling#

7.1. Characteristics of the mesh#

We discretize the structure into 5 finite elements QUAD4.

The nodes on either side of the interface are enriched nodes, so the three central cells with such nodes are also enriched. Only the two extreme stitches are classic knits with only classic knots.

It is therefore possible to impose boundary conditions on the extreme meshes in the usual way.

_images/10000000000000C8000001F431D73E167571B76C.png

Figure 7.1-a : F mesh

7.2. Boundary conditions#

The boundary conditions applied represent the same physical phenomenon as for modeling A. The nodes on the lower face are embedded and a displacement of the nodes on the upper face is imposed:

Underside (Knots \(\mathit{N1}\) and \(\mathit{N2}\)): \(\mathrm{DX}=0\) and \(\mathrm{DY}=0\)

Upper side (Nodes \(\mathit{N3}\) and \(\mathit{N4}\)): \(\mathrm{DX}=0\), \(\mathrm{DY}=\mathrm{uy}={10}^{-5}\).

7.3. Analytical resolution#

The solution of such a problem is of course still obvious: all the movements following \(x\) are zero, all the movements following \(y\) below the level set are zero and all the movements following \(y\) above the level set are equal to the displacement imposed \({u}_{y}\) at the top of the structure.

7.4. Tested sizes and results#

The displacement values are tested after convergence of the iterations of the operator STAT_NON_LINE. We check that we find the values determined in [§ 7.3].

Identification

Reference

Tolerance

DX for all nodes just below the interface

0.00

1.0E-16

DY for all nodes just below the interface

0.00

1.0E-16

DX for all nodes just above the interface

0.00

1.0E-16

DY for all nodes just above the interface

1.0E-6

1.0E -9%

To test all the nodes at once, we test the MINIMUM and the MAXIMUM of the column.

We also test the values of the displacement from the command POST_CHAM_XFEM. We are in fact testing the value of the sum of the absolute values of the movements of the nodes of the cracked mesh. It is a non-regression test compared to the values obtained with version 8.2.13 for \(\mathit{DX}\) and 9.0.21 for \(\mathit{DY}\).

Identification

Reference

Difference

SOMM_ABS (DX)

0.000

1.E-12

SOMM_ABS (DY)

1.3E-05

1.0E -04%

7.5. Comments#

We notice the discontinuity of the field of movement when crossing the interface, which is possible thanks to the enrichment of the elements with the Heaviside degree of freedom.