7. E modeling#
7.1. Characteristics of modeling#
It’s a \(\mathrm{3D}\) XFEM modeling with linear elements.
This modeling allows a direct comparison between 2 preconditioning methods: the elimination of Heaviside ddls with the stiffness criterion (modeling \(D\)) and the pre-conditioner XFEM (modeling \(E\)). As in modeling \(D\), we place ourselves just before the interface readjustment threshold, at 1.1% of the length of the edge. In command MODI_MODELE_XFEM, we pass the option PRETRAITEMENTS =” FORCE “[U4.41.11], which activates the pre-conditioner XFEM. The use of pre-conditioner XFEM makes it possible to gain 3 orders of magnitude in terms of the precision of the results.
The equation for interface XFEM is: \(x+y+z+0.011=0\). The preconditioner is activated by PRETRAITEMENTS =” FORCE “[U4.41.11] for this modeling. The ddls Heaviside elimination criterion is therefore not activated.
7.2. Characteristics of the mesh#
Same mesh as modeling \(\text{A}\).
7.3. Tested sizes and results#
We test the maximum error on the displacement, in absolute value, along the interface XFEM. A list of points located on the interface is extracted, as well as the displacement field interpolated at these points.
As the displacement field is discontinuous, at each point of the interface, there are 2 analytical values of the displacement field \({U}^{\text{+}}\left(x,y,z\right)=\{{U}_{1}^{\text{+}},{U}_{2}^{\text{+}},{U}_{3}^{\text{+}}\}\) and \({U}^{\text{-}}\left(x,y,z\right)=\{{U}_{1}^{\text{-}},{U}_{2}^{\text{-}},{U}_{3}^{\text{-}}\}\). These analytical values are compared to the interpolated displacement field at each point.
In practice, in the Code_Aster to take into account the discontinuity during interpolation, each point on the interface is transformed into duplicate nodes (NP*and NM*) to which are associated displacement values \({\mathit{DX}}_{i}(\mathit{NP})\) and \({\mathit{DX}}_{i}(\mathit{NM})\).
For the « PLUS » nodes (noted NP by default in the Code_aster), the following difference table is then calculated \(\mathit{DIFF}{(\mathit{NP})}_{i}=∣{U}_{i}^{\text{+}}({x}_{\mathit{NP}},{y}_{\mathit{NP}},{z}_{\mathit{NP}})-{\mathit{DX}}_{i}(\mathit{NP})∣\);
for the nodes « MOINS » (noted NM by default in the Code_aster), the following difference table is then calculated \(\mathit{DIFF}{(\mathit{NM})}_{i}=∣{U}_{i}^{\text{-}}({x}_{\mathit{NM}},{y}_{\mathit{NM}},{z}_{\mathit{NM}})-{\mathit{DX}}_{i}(\mathit{NM})∣\).
Identification |
Reference |
Type |
% tolerance |
\(\text{DIFF}{\text{(NP)}}_{X}\) (MAX) |
0.0 |
Analytics |
1.E-09 |
\(\text{DIFF}{\text{(NP)}}_{Y}\) (MAX) |
0.0 |
Analytics |
1.E-09 |
\(\text{DIFF}{\text{(NP)}}_{Z}\) (MAX) |
0.0 |
Analytics |
1.E-09 |
\(\text{DIFF}{\text{(NM)}}_{X}\) (MAX) |
0.0 |
Analytics |
1.E-03 |
\(\text{DIFF}{\text{(NM)}}_{Y}\) (MAX) |
0.0 |
Analytics |
1.E-03 |
\(\text{DIFF}{\text{(NM)}}_{Z}\) (MAX) |
0.0 |
Analytics |
1.E-03 |
Table 7.3-1 : summary results
7.4. note#
We gain several orders of magnitude in the precision of the results, compared to modeling \(\text{D}\), for the same position of the interface. In modeling \(\text{E}\), the conditioning is controlled by the pre-conditioner and of the same order of magnitude as for modeling \(\text{D}\) (around \({10}^{3}\)).