4. B modeling

4.1. Characteristics of modeling

In this modeling, still D_ PLAN, we consider the case X- FEM. The crack is no longer meshed, it is introduced into the healthy mesh by the operator DEFI_FISS_XFEM.

4.2. Characteristics of the mesh

The structure is modelled by a regular mesh composed of \(30\times 50\) QUAD4, respectively along the \(x,y\) axes (see [Figure 4.2‑a]). It can be observed that the cells affected by the crack are partitioned into triangles by the X- FEM operators for the purposes of numerical integration of quantities such as mass and stiffness.

Figure 4.2‑a: Meshing for modeling B

4.3. Tested features

The first 8 natural modes of the structure are calculated by considering the contribution of the geometric rigidity due to the prestress induced to equilibrium by the load in question.

Orders
CALC_MATR_ELEM		OPTION = “RIGI_MECA” OPTION = “MASS_MECA” OPTION = “RIGI_GEOM”
CALC_MODES	CALC_FREQ SOLVEUR_MODAL	OPTION = “PLUS_PETITE” NMAX_FREQ = 8 METHODE = “SORENSEN”
POST_CHAM_XFEM		RESULTAT = sd_mode_meca

4.4. Tested sizes and results

For this modeling, tests on the natural frequencies of the first 8 modes are considered with the results obtained from modeling A.

Identification	Reference	Code_Aster	% difference
Frequency mode 1	7.005	6.902	1.5
Frequency mode 2	24.895	24.895	24.984	0.36
Frequency mode 3	41.820	41.143	1.6
Frequency mode 4	84.905	84.905	84.661	0.29
Frequency mode 5	106.179	106.179	103.269	2.7
Frequency mode 6	134.298	134.298	135.242	0.70
Frequency mode 7	166.198	165.284	0.55
Frequency mode 8	181.048	181.048	181.556	0.28

4.5. notes

As can be observed in the comparison of the results for this modeling, differences are obtained between the natural frequencies calculated with the X- FEM model and those calculated with the classical model. These differences are normal knowing that the mass, the elastic stiffness as well as the geometric contribution of the stiffness are calculated differently for the elements X- FEM. These are partitioned into triangles on which 12-point integration schemes are considered. On the other hand, for the classical elements the 4-point diagram is used. For some specific modes (in particular mode 5 here) the difference is even greater because in this case the contribution of geometric rigidity plays a more important role. As for the reference solution the mesh is quite coarse around the tip of the crack, the stress field is not calculated with the same precision as in the case X- FEM where special enrichment functions allow a more accurate evaluation. With regard to modal deformations, we note (see Figure 4.5-a) a very good agreement between the results obtained from the classical calculation and those obtained from the calculation X- FEM.

mode 1*mode 2**mode 3 mode 4 mode 5

Figure 4 .5 - a: The modal deformations for the first 5 natural modes. On the top row are the « classic » modes and on the bottom row are the « X- FEM » modes