D modeling ============== Characteristics of modeling ------------------------ It's a :math:`\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 :math:`D`) and the pre-conditioner XFEM (cf. modeling :math:`E`). We place ourselves just before the interface readjustment threshold, at 1.1% of the length of the edge. In command MODI_MODELE_XFEM, we leave the option PRETRAITEMENTS =' AUTO '[:ref:`U4.41.11 `], which deactivates the XFEM preconditioner for linear elements. The equation for interface XFEM is: :math:`x+y+z+0.011=0`. The preconditioner is not activated in PRETRAITEMENTS =' AUTO '[U4.41.11] for this modeling. The ddls Heaviside elimination criterion is therefore activated. Characteristics of the mesh ---------------------------- Same mesh as modeling :math:`\text{A}`. 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 :math:`{U}^{\text{+}}\left(x,y,z\right)=\{{U}_{1}^{\text{+}},{U}_{2}^{\text{+}},{U}_{3}^{\text{+}}\}` and :math:`{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 :math:`{\mathit{DX}}_{i}(\mathit{NP})` and :math:`{\mathit{DX}}_{i}(\mathit{NM})`. *For the "PLUS" nodes (noted NP* by default in the Code_aster), the following difference table is then calculated :math:`\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 :math:`\mathit{DIFF}{(\mathit{NM})}_{i}=∣{U}_{i}^{\text{-}}({x}_{\mathit{NM}},{y}_{\mathit{NM}},{z}_{\mathit{NM}})-{\mathit{DX}}_{i}(\mathit{NM})∣`. .. csv-table:: "**Identification**", "**Reference**", "**Type**", "**% tolerance**" ":math:`\text{DIFF}{\text{(NP)}}_{X}` (MAX)", "0.0", "Analytics", "1.E-07" ":math:`\text{DIFF}{\text{(NP)}}_{Y}` (MAX)", "0.0", "Analytics", "1.E-07" ":math:`\text{DIFF}{\text{(NP)}}_{Z}` (MAX)", "0.0", "Analytics", "1.E-07" ":math:`\text{DIFF}{\text{(NM)}}_{X}` (MAX)", "0.0", "Analytics", "1.E-01" ":math:`\text{DIFF}{\text{(NM)}}_{Y}` (MAX)", "0.0", "Analytics", "1.E-01" ":math:`\text{DIFF}{\text{(NM)}}_{Z}` (MAX)", "0.0", "Analytics", "1.E-01" **Table** 6.3-1 **: summary results** note -------- We gain several orders of magnitude in the precision of the results, compared to modeling :math:`\text{A}`, knowing that, the interface has been offset by :math:`0.1\text{\%}`. The readjustment of the interface (with a threshold of 1%) therefore has a harmful influence on the quality of the results. However, to control the conditioning in modeling :math:`\text{D}`, we note the activation of the elimination of ddls Heavisides, thanks to a criterion for estimating rigor [:ref:`R7.02.12 `].