# Numerical Modeling Technique of Damage Behavior of
## 2 Introduction of XFEM
### Numerical Integration Strategy  
In this study, an efficient numerical integration strategy is adopted to accurately assemble the contribution to the weak form on both sides of the crack surface. This strategy is simple yet effective, and it is also capable of adapting to much more complex crack-propagation problems. The elements cut by the crack are firstly subdivided into 16 sub-quadrilaterals, as shown in Figure 2. Following this, two sets of subdivision strategies are adopted to remedy the integration error due to the incompatibility between the sub-quadrilateral edges and the crack surface [51]. To be specific, one set with sub-elements compatible with the crack surface is used to compute the global stiffness matrix, as shown in Figure 2a,b, and the other set with sub-elements independent of the crack surface is used for residual vector and plastic flow, as denoted in Figure 2c,d. For each sub-element, four-order standard Gauss-Legendre quadrature is used to determine the location of integration points. In the first set of the sub-element division strategy (as shown in Figure 2a,b), sub-quadrilaterals that cracks pass through are further divided into several sub-blocks according to the intersection of the crack surface with the sub-quadrilateral. Considering that the crack propagation occurs in an arbitrary manner in a finite element solution domain during the analysis, there should be a series of intersection situations that may occur. In the following, the possible intersection situations are summarized in detail, and the division strategy of sub-quadrilaterals for each case is elaborated. In Figure 3, the subdivision strategy of the sub-quadrilaterals with a crack tip is presented. First of all, the subdivision for three simple intersection situations (the crack tip touches the boundary of a sub-quadrilateral but not penetrate it) is shown in Figure 3a-c.  
A sub-quadrilateral is subdivided into four to six triangles with the crack tip as a common vertex if the crack tip lies in the sub-quadrilateral but not on the boundary, as illustrated in Figure 3d-g. In order to integrate the function on a triangle within a prescribed accuracy, a mapping scheme with a two-step coordinate transform is adopted in this study, as depicted in Figure 4. Each triangle is firstly mapped on a reference isosceles right-angled triangle; then, this reference triangle is further mapped on a reference square with a side of length 2 by Duffy coordinate transform [52]. A sub-quadrilateral is further divided into a set of quadrilaterals; if it is penetrated by a straight crack and the crack tip lies on the boundary, see Figure 3h-l. That is, a triangle is further divided into three sub-quadrilaterals by connecting the midpoint of each side and the centroid of the triangle. Figure 2: Element partitioning strategy. For a sub-quadrilateral that is penetrated by the crack but does not contain the crack tip, the subdivision strategy is illustrated in Figure 5. Similar to crack tip cases, two simple division situations without an actual division needed are firstly presented in Figure 5a,b. In Figure 5c-e, the sub-quadrilaterals containing a crack tip before (Figure 3d-g) have been cut through by the propagating crack. Similar to the division strategy shown in Figure 3d-g, these sub-quadrilaterals are also divided into a set of triangles for consistency, and the same mapping scheme is also adopted for each triangle. Following this, a sub-quadrilateral penetrated by a straight crack is divided into several quadrilaterals (see Figure 5f-i) in the same manner as the division strategy shown in Figure 3h-l.  
Figure 4: Mapping scheme. Figure 3: Subdivision of sub-quadrilaterals with a crack tip. With the above strategy, sub-quadrilaterals cut by the cracks are divided into several simple geometric figures (triangles and quadrilaterals), and the location and number of integration points are then determined.