## 3 Extended Finite Element Method: Review
### Incorporating Discontinuities into Finite Element Spaces  
Consider a domain \(\Omega\) which is crossed by a discontinuity surface \(\Gamma_{d}\), as illustrated in Figure 1 (for simplicity we show a body which is completely crossed by a discontinuity surface, but it also possible to consider bodies which are not bisected by a discontinuity surface, and we will do so for the examples presented in Section 5.). The domains on different sides of the discontinuity surface are denoted by \(\Omega^{-}\) and \(\Omega^{+}\) such that \(\Omega^{-}\bigcup\Omega^{+}\bigcup\Gamma_{d}=\Omega\). The unit normal vector \(\mathbf{n}\) on \(\Gamma_{d}\) is defined such that it points towards \(\Omega^{+}\), as illustrated in Figure 1.  
We wish to represent functions which are continuous on \(\Omega\backslash\Gamma_{d}\) and exhibit jumps across the surface \(\Gamma_{d}\). Such a function \(u\) can be expressed in terms of continuous functions and a jump function by  
\[u=\bar{u}+H_{d}\hat{u}, \tag{5}\]  
where \(\bar{u}\) and \(\hat{u}\) are defined on \(\Omega\) and are continuous, and \(H_{d}\) is the Heaviside function centred on the discontinuity surface \(\Gamma_{d}\) and is defined such that \(H_{d}\left(\mathbf{x}\right)=1\) if \(\mathbf{x}\in\Omega^{+}\) and \(H_{d}\left(\mathbf{x}\right)=0\) if \(\mathbf{x}\in\Omega^{-}\). The jump in \(u\) across the surface \(\Gamma_{d}\), denoted by \(\llbracket u\rrbracket=u^{+}-u^{-}\), and is equal to \(\hat{u}\) for \(\mathbf{x}\in\Gamma_{d}\). The jump may or may not be required to go to zero on \(\partial\Gamma_{d}\). In the context of a crack, \(\partial\Gamma_{d}\) is the crack tip.  
The Heaviside function can be built into a finite element basis by exploiting the partition-of-unity property of the finite element shape functions. A regular finite element function \(u_{h}\) is expressed in terms of the finite element shape functions \(\{\phi_{i}\}\) and the nodal degrees of freedom \(\{\bar{u}_{i}\}\),  
\[u_{h}=\sum_{i}\phi_{i}\bar{u}_{i}. \tag{6}\]  
A finite element function which exhibits a jump across the surface \(\Gamma_{d}\) can be represented by adding the Heaviside function to the finite element basis,  
\[u_{h}=\sum_{i}\phi_{i}\bar{u}_{i}+\sum_{i}H_{d}\phi_{i}\hat{u}_{i} \tag{7}\]  
where \(\hat{u}_{i}\) are \"enriched\" degrees of freedom and are associated with the field \(\hat{u}\) in Equation (5). The expression in (7) will in general lead to a linear dependency if used in a finite element formulation, since away from a discontinuity surface the Heaviside function is a constant over a finite element cell, and the regular finite element basis contains constant functions. This linear dependency can be obviated as follows: enriched degrees of freedom \(\hat{u}_{i}\) at node \(i\) are active only if the support of the shape function \(\phi_{i}\) is intersected by a discontinuity surface. Formally,  
\[u_{h}=\sum_{i}\phi_{i}u_{i}+\sum_{i}\left\{\psi\right\}_{i}\phi_{i}\hat{u}_{i} \tag{8}\]  
where \(\left\{\psi\right\}_{i}=H_{d}\) if the support of \(\phi_{i}\) is intersected by a discontinuity surface, otherwise \(\left\{\psi\right\}_{i}=\emptyset\). The extra degrees of freedom are therefore localised to the small region around the discontinuity surface and eliminated elsewhere.