XFEM for Crack Modelling
4.4 Numerical Aspects
4.5.3 Convergence Analysis In this section, five methods are compared:In this section, five methods are compared:
• The finite element method (FEM), Section 4.2.
2 3 4 5 6 7 8 9 0
Δa= 0.5 cm Mesh 1
Δa= 0.5 cm Mesh 2
Δa= 0.4 cm Mesh 2
Figure 4.40: Zoom-in of the crack paths for different mesh-Δacombinations.
σ−criterion
S−criterion
Figure 4.41: Zoom-in of the crack paths for theσ−criterion and theS−criterion, considering the second (fine) mesh andΔa= 0.5 cm.
• The classical eXtended Finite Element Method, where only the nodes whose support contains the crack tip are enriched with near-tip functions (XFEM), Section 4.3.3.
• XFEM with near-tip enrichment on a fixed area (XFEM-FA), Section 4.3.7.
• XFEM considering the gathering of the degrees of freedom associated to the near-tip enrichment (XFEM-FA-Gath), Section 4.3.8.
• XFEM using a cut-off function (XFEM-Cutoff), Section 4.3.9.
0 20 40 60 80 100 120 140 σ√a0
KIC
u2E KIC√a0 0.05
0.10 0.15 0.20 0.25 0.30 0.35
σ−criterion S−criterion
Figure 4.42: Load-displacement curves for theσ−criterion and theS−criterion, con-sidering the second (fine) mesh andΔa= 0.5 cm.
For each method, the convergence rater, in terms of the mesh sizeh, is studied for the following parameters:
• L2-norm over the entire domain and over a subdomain not containing the crack tip.
• Energy norm over the entire domain and over a subdomain not containing the crack tip.
• Stress intensity factors.
• T-stress.
The tests are carried out considering LAGRANGEfinite elements of orderp= 1, 2 and3. The domainΩis the unit square[0,1]×[0,1] (in meters) and the crack is defined by the segment going from(0,0.5)to(0.5,0.5). Different regular meshes are obtained by dividing the unit square. Divisions going from 5 to 75 (in both directions) are considered, see Figure 4.5.3 for an example mesh. The mesh sizehis chosen as the side length of the elements.
In the case of the XFEM-FA and XFEM-FA-Gath a value ofamax= 0.1 mis chosen, while for the XFEM-Cutoff the valuesr1= 0.1 mandr2= 0.4 mare considered.
TheL2-norm, energy norm and stress intensity factors are calculated for the solu-tion obtained by imposing the asymptotic displacement field (2.4) withKI=KII = 1 N/mm3/2 as DIRICHLET boundary condition, while for the T-stress the values KI =KII = 10 N/mm3/2andT = 1 N/mm2 are used. Considering that the exact solution of the problem does not involve terms of orderr3/2or higher, only one set of near-tip functions is added in the XFEM approximation.
Since the HEAVISIDEenrichment is active at the DIRICHLETboundaryΓu, theknown nodal degrees of freedom at the boundary are calculated by minimizing
u∗−uhL2(Γu)=
Γu
u∗−uh2dΓ. (4.107)
The subdomain considered for theL2-norm and energy norm is the square[0.6,0.8]× [0.6,0.8]in the case of the XFEM methods, while for the FEM the subdomain is given by[0.625,0.875]×[0.625,0.875].
In the case of the stress intensity factors andT-stress, they are extracted from the numerical solution using the domain interaction integralJ(1,2)and appropriate auxil-iary solutions, as described in sections 2.9 and 2.10. The domain over whichJ(1,2) is evaluated, is chosen as composed of all the elements with a node within a circle of radius0.2maround the crack tip. Thus, this domain depends only weakly on the mesh size.
In all cases, the stiffness matrix of the element containing the crack tip was obtained using the integration technique proposed in [Laborde et al., 2005], where the inte-gration points of the sub-triangles with vertex on the crack tip are obtained from the unit square via mapping.
The results of the convergence test are summarized in Table 4.2. Figure 4.44 presents the convergence curves for the energy-norm in the entire domain, con-sidering only the FEM and the classical XFEM. It can be seen that the convergence rate is independent of the polynomial degree and is restricted to1/2. Figure 4.45 shows the convergence curves for the stress intensity factorKII, while Figure 4.46 shows the condition number of the stiffness matrixκfor the different methods. From these results some remarks are worth mentioning:
• The finite element method shows a convergence rate equal to1for all the stud-ied parameters, with exception of the energy norm in the entire domain, where a poor convergence rater = 1/2is achieved. These results are independent of the order of the finite element basisp. The classical XFEM presents the same behaviour, the only surprising result was a convergence rater≈pfor the energy norm in a subdomain not containing the crack tip. This result needs further investigation.
• The methods XFEM-FA and XFEM-Cutoff show a similar behaviour, with con-vergence ratesr≈p+ 1in theL2-norm andr≈pin the energy norm. In the case of the stress intensity factors andT-stress, a convergence rater≈2pis achieved. For the XFEM-FA withp= 3it is difficult to recognize a convergence rate in the case of the stress intensity factors andT-stress, probably due to the fact that the accuracy of the method is comparable with the machine precision and that the stiffness matrix becomes very sensitive due to its poor condition-ing. Actually, the condition number forp = 3 is of orderO(h−8.7), while the XFEM-Cutoff shows an orderO(h−3.0).
• The method XFEM-FA-Gath shows a lost of convergence with respect the op-timal convergence rates, but the convergence rates increase with the order of
the finite element basis. In the energy norm over the entire domain, the method shows a lost of half an order (independent ofp). The same occurs for theL2 -norm over the entire domain with exception of the case p = 1, where one order was lost. In the case of the SIFs andT-stress, one order was lost in the convergence rate (again independent ofp), with respect to the XFEM-FA and XFEM-Cutoff methods. This lost of convergence is attributed to the blending elements, see [Laborde et al., 2005].
u=u∗(KI= 1, KII= 1)
L
L
Figure 4.43: Example mesh used for the analysis.