*2.2. Discretization of Governing Equations*

In the XFEM, the displacement discontinuity can be directly embedded by introducing additional degrees of freedom onto existing nodes whose supports are intersected by discontinuities. Comprehensive overviews of the XFEM have been given by numerous studies [35–37].

The generalized form of the XFEM approximation of the displacement field can be written as

$$u^h(\mathbf{x}) = \sum\_{A} N\_i(\mathbf{x}) \cdot u\_i + \sum\_{J} N\_j(\mathbf{x}) \cdot \phi\_j(\mathbf{x}) \cdot a\_j \tag{3}$$

In the above function, the standard FE approximation ∑ *A Ni*(*x*)·*ui* represents the continuous part of the displacement field, while the second term represents the discontinuous

part, where *ui* and *aj* are standard and enriched DOFs, respectively, *φj*(*x*) are the enrichment functions, which take different forms for specific kinds of discontinuity problems, A denotes the set of all nodes, and *J* denotes the pre-selected set of local nodes associated with discontinuities.

The weak form of the governing partial differential equation can be derived from the principle of virtual work and the Galerkin procedure. When the cohesive crack model is assumed, the cohesive traction fc that transferred is a function of the crack opening ω. The weak form of the equilibrium equation can be expressed as:

$$\mathcal{W}^{int} = \mathcal{W}^{ext} + \mathcal{W}^{coh} \tag{4}$$

Or

$$\int\_{\Omega} \boldsymbol{\sigma} \cdot \boldsymbol{\delta\varepsilon} \, d\Omega = \int\_{\Omega} \boldsymbol{b} \cdot \boldsymbol{\delta u} \, d\Omega + \int\_{\Gamma\_{t}} \overline{\boldsymbol{t}} \cdot \boldsymbol{\delta u} \, d\Gamma + \int\_{\Gamma\_{t}} \boldsymbol{f}^{c} \cdot (\boldsymbol{\delta u}^{+} - \boldsymbol{\delta u}^{-}) \, d\Gamma \tag{5}$$

Discretization of Equation (5) in the XFEM framework results in:

$$Kq = \lambda f^{\text{ext}} + f^{\text{coh}} \tag{6}$$

where *q* is the generalized nodal displacement vector, *q<sup>e</sup>* = [ *u<sup>e</sup> <sup>i</sup> <sup>a</sup><sup>e</sup> <sup>i</sup>* ] for each element, and λ is the load factor.

The stiffness matrix *K* is composed of

$$K = \begin{bmatrix} K^{uu} & K^{ua} \\ K^{ua} & K^{ua} \end{bmatrix} \tag{7}$$

With

$$\begin{array}{c} \mathcal{K}^{uu} = \int\_{\Omega} \left( \boldsymbol{B}^{u} \right)^{T} \boldsymbol{D} \boldsymbol{B}^{u} \boldsymbol{d} \Omega \\\ \mathcal{K}^{ua} = \int\_{\Omega\_{\text{var}}} \left( \boldsymbol{B}^{a} \right)^{T} \boldsymbol{D} \boldsymbol{B}^{u} \boldsymbol{d} \Omega \\\ \mathcal{K}^{aa} = \int\_{\Omega\_{\text{var}}} \left( \boldsymbol{B}^{a} \right)^{T} \boldsymbol{D} \boldsymbol{B}^{a} \boldsymbol{d} \Omega + \int\_{\Gamma\_{c}} \boldsymbol{N}^{T} \boldsymbol{T}^{c} \boldsymbol{N} \boldsymbol{d} \Gamma \end{array} \tag{8}$$

where *T*c is the tangential modulus matrix of the cohesive crack determined by the cohesive crack behavior and is obtained from the relation *<sup>T</sup><sup>c</sup>* <sup>=</sup> *<sup>∂</sup> <sup>f</sup> coh*(*ω*) *∂ω* . The external nodal force *f* ext and the cohesive nodal force *f* coh can be obtained as

$$\begin{cases} f^{\text{ext}} = \lambda \int\_{\Gamma\_{\text{f}}} N^T \mathbf{\tilde{f}} \mathbf{d} \Gamma + \int\_{\Omega} N^T b \mathbf{d} \Omega\\ f^{\text{coh}} = -\int\_{\Gamma\_{\text{c}}} \sigma\_{\mathcal{Y}}(\omega) (N^T\_+ - N^T\_-) \, \mathbf{d} \Gamma \end{cases} \tag{9}$$

where the crack opening displacement ω can be given by

$$
\omega = \stackrel{\rightarrow}{n} \cdot (u^+ - u^-) = \stackrel{\rightarrow}{n} \cdot 2 \sum\_{i} N\_i a\_i \tag{10}
$$

It can be observed from the Equations (8) and (9) that the cohesive behavior has a direct effect on both the stiffness matrix *K* and the nodal force vector *f* coh. The relation between the cohesive force and the crack opening makes the problem nonlinear.

The four enrichment schemes we examined are designed to consider the effect of their variations, including the employment of tip branch functions and a corrected approximation in blending elements. These four schemes are denoted XFEM-h, XFEM-s, XFEM-c1, and XFEM-c2, and detailed as follows.
