*2.3. XFEM-h*

Because the singularity of the displacement field around the crack tip vanishes, a heaviside function is suitable for the entire crack, including the crack tip. In this scheme, the approximation of the displacement field can be written as

$$u^h(\mathbf{x}) = \sum\_{i \in A} N\_i(\mathbf{x}) \cdot u\_i + \sum\_{j \in J} N\_j(\mathbf{x}) \cdot \left[ H(\mathbf{x}) - H(\mathbf{x}\_j) \right] \cdot a\_j \tag{11}$$

where *H*(*x*) is the heaviside function, which takes +1 on one side of the crack and −1 on the other side, and *J* is the set of nodes whose supports are fully cut by the crack, which is depicted in Figure 3a.

**Figure 3.** Node subsets and element types in different enrichment schemes. (**a**) XFEM-h; (**b**) XFEM-s; (**c**) XFEM-c1; and XFEM-c2.

In order to make the displacement gap vanish to zero at the crack tip within the tip element, we extended the method proposed by Zi and Belytschko [29] to quadrilateral elements. Specifically, for the tip element, the modified shape function *Nj*(*x*) was used instead of the standard shape function *Nj*(*x*). As shown in Figure 4, if the crack intersects with boundary 14 within the tip element, we make a straight line through the crack tip point and intersect the element boundary at points 5 and 6. Then, the shape function *Nj*(*x*) used for the tip element is actually the standard shape function of virtual element 1564. Since nodes 1 and 4 are enriched, the discontinuous part of the displacement can be written as

$$u\_{\rm disc} = a\_1 \overline{N}\_1(\mathbf{x}^\*) [H(\mathbf{x}^\*) - H(\mathbf{x}\_1)] + a\_4 \overline{N}\_4(\mathbf{x}^\*) [H(\mathbf{x}^\*) - H(\mathbf{x}\_4)] \tag{12}$$

where *x*\* are the coordinates of virtual element 1564.

**Figure 4.** Approximation of the displacement field in the tip element. (**a**) The crack intersecting with boundary 14; (**b**) the corresponding parent element.

Since this scheme treats the entire domain with the heaviside function only, the blending with the unenriched subdomain does not occur, which implies that the PU holds in the entire domain.
