## 3 The Extended/Generalized Finite Element Methods (XFEM/GFEM)
### Crack Propagation
#### 3.7.1 Stress Intensity Factors  
One of the most widely used techniques for the extraction of SIFs in extended, generalized or standard finite element simulations, involves the use of the interaction integral. This can be derived by initially converting the \(J\) integral in a domain form and subsequently evaluating it for a stress state resulting from the superposition of an auxiliary stress state and the computed numerical solution. Then the interaction term of the integral, for two dimensional problems, assumes the form:  
\[I=-\int_{V}q_{j}\left(\sigma_{kl}\epsilon^{\rm aux}_{kl}\delta_{1j}-\sigma^{ \rm aux}_{kj}u_{k,1}-\sigma_{kj}u^{\rm aux}_{k,1}\right)\,dV \tag{26}\]  
where \(\epsilon^{\rm aux}\), \(\sigma^{\rm aux}\) and \(u^{\rm aux}\) are the auxiliary stress, strain and displacement fields respectively which can be defined as in Moes et al. [44] and \(q\) is a virtual velocity field. Typically, \(q\) is chosen to assume a value of one for nodes within a disc of radius \(r_{d}\) around the crack tip and a value of zero for the remaining nodes.  
In the interior of the elements, the values of \(q\) are interpolated using the FE basis functions. As a result, the expression of Equation (26) needs to be evaluated only in a \"ring\" or layer of elements around the crack tip. The components of the tensors of Equation (26), refer to a basis aligned with the crack, which for implicit crack representations can be conveniently defined using the level sets [136; 137]. By considering the relation between the \(J\) integral and the SIFs it is straightforward to show that with an appropriate selection of the SIF values of the auxiliary state, the SIFs can be directly obtained from the interaction integral.  
It should be noted that in the derivation of Equation (26) it has been assumed that the crack is straight. Of course, the expression can also be used for curved cracks, perhaps with some loss of accuracy, provided that the curvature of the crack is not very pronounced within the interaction integral domain. Alternatively, a more complicated formulation can be used [154], leading to more accurate results.  
For three dimensional problems a more complicated expression for the interaction integral needs to be used as, for instance, in Gosz and Moran [15]. Furthermore, different alternatives exist for the definition of the virtual velocity field and the domain of integration [155; 132; 152] as well as the basis on which the tensor components refer to [154; 155].  
Alternative methods of SIF extraction, employed in the XFEM/GFEM context, include direct extraction based on the enriched degree of freedom values [108; 109; 110; 114], Irwin's integral [156; 157; 158; 159], and extraction through crack opening displacements [160]. The former technique relies on the fact that when vector enrichment is used, the physical meaning of the enriched degrees of freedom corresponding to the tip enriched nodes is by definition equivalent to the SIFs.  
In several works [108; 109; 110], the technique is combined to degree of freedom gathering and the use of higher order terms of the Williams expansion to increase the accuracy of the extracted SIFs. Similarly, extraction using Irwin's integral also requires higher order enrichment. A relative advantage of both of these methods is their low computational cost and the fact that they do not require the use of auxiliary fields as in the interaction integral method. Extraction through crack opening displacements [160] does not require the use of higher order enrichment functions and is computationally inexpensive, it does however employ auxiliary fields. Finally, it should be mentioned that even though some of the above methods might be advantageous for certain problems, the interaction integral method is in general more accurate and has in general a wider field of applicability since domain integral formulations are available also for problems outside the LEFM domain.