5. C modeling#
5.1. Characteristics of modeling#
It’s a \(\mathrm{3D}\) XFEM modeling with quadratic elements.
The interface is positioned exactly as in modeling \(A\) (at 1% of the edge). Since the estimation criteria are not adapted to the quadratic elements, only the preconditioner controls the conditioning of the stiffness matrix. The test shows that the preconditioner makes it possible, on the one hand, to position the interface XFEM, completely independent of the proximity of the nodes of the mesh and, on the other hand, considerably increases the precision of the results compared to the \(A\) modeling.
The equation for interface XFEM is: \(x+y+z+0.01\mathrm{=}0\), in the same configuration as the \(A\) modeling.
5.2. Characteristics of the mesh#
Same mesh as modeling \(B\).
5.3. 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 displacement field interpolated at each point by ASTER 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 5.3-1 : summary results