## 4 The Scaled Boundary Finite Element Method (SBFEM)
### Calculation of SIFs  
Since the general solution to the SBFEM equation is extracted as a power series, the singular modes are readily identified: By inspection of \(\mathbf{S_{i}}\) any \(-1<real(\lambda)<0\) will result in a singularity at \(\xi=0\). Placement of the scaling center at a crack tip may be exploited to calculate the generalized SIFs (Figure 2a). By including a double node at the crack mouth, two additional modes, the singular modes, arise (Figure 3), whose eigen-vectors resemble the mode I and mode II fracture cases. The singular stress field is extracted from the general solution (Equation (42)), where the superscript \({}^{(s)}\) denotes the singular quantities:  
\[\mathbf{\sigma^{(s)}(\xi,\eta)}=\mathbf{\Psi}_{\sigma}^{(s)}(\eta)\xi^{-S^{(s)}-I}\mathbf{ c^{(s)}} \tag{44}\]  
For consistency with other numerical methods and experimental reporting, a characteristic length \(L\) is introduced and a transformation to polar coordinates is sought (Figure 2b):  
\[\xi=\frac{r}{r_{b}(\theta)}=\frac{L}{r_{b}(\theta)}\times\frac{r}{L} \tag{45}\]  
The singular stress field is equivalently expressed in polar coordinates as:  
\[\mathbf{\sigma^{(s)}(r,\theta)}=\mathbf{\Psi}_{L}^{(s)}(\theta)(\frac{r}{L})^{-S^{(s) }-I}\mathbf{c^{(s)}} \tag{46}\]  
implying the corresponding stress modes \(\mathbf{\Psi}_{L}^{(s)}(\theta)\) given by:  
\[\mathbf{\Psi}_{L}^{(s)}(\theta)=\mathbf{\Psi}_{\sigma}^{(s)}(\eta(\theta))(\frac{L}{ r_{b}(\theta)})^{-S^{(s)}-I} \tag{47}\]  
For the case of 2D elastostatics, two singular stress modes exist. Hence, \(\mathbf{S^{(s)}}\) and \(\mathbf{\Psi}_{L}^{(s)}(\theta)\) reduce to matrices of size \((2\times 2)\), while both \(\mathbf{c^{(s)}}\) and \(\mathbf{\sigma^{(s)}(r,\theta)}\) form vectors of size \((2\times 1)\). More specifically, only the components of \(\mathbf{\sigma^{(s)}(r,\theta)}=(\sigma_{\theta}^{(s)}(r,\theta),\tau_{\sigma\theta }^{(s)}(r,\theta))^{T}\) are retained, which correspond to mode I and II cracks, for which the formal definition of the gSIFs at angle \(\theta\) is given as [50]:  
\[\begin{cases}\sigma_{\theta}^{(s)}(r,\theta)\\ \tau_{r\theta}^{(s)}(r,\theta)\end{cases}=\frac{1}{\sqrt{2\pi L}}(\frac{r}{L}) ^{-\mathbf{S^{(s)}}(\theta)}\begin{cases}K_{I}(\theta)\\ K_{II}(\theta)\end{cases} \tag{48}\]  
The _matrix of orders of singularity \(\mathbf{\tilde{S}^{(s)}(\theta)}\)_ is introduced such that:  
\[\mathbf{\tilde{S}^{(s)}(\theta)}=\mathbf{\Psi}_{L}^{(s)}(\theta)(\mathbf{S^{(s)}+I})\mathbf{ \Psi}_{L}^{(s)}(\theta)^{-1} \tag{49}\]  
\[\begin{cases}\sigma_{\theta}^{(s)}(r,\theta)\\ \tau_{r\theta}^{(s)}(r,\theta)\end{cases}=(\frac{r}{L})^{-\mathbf{S^{(s)}}(\theta )}\mathbf{\Psi}_{L}^{(s)}(\theta)\mathbf{c^{(s)}} \tag{50}\]  
Comparing Equation (48) with Equation (50) permits the evaluation of the gSIFs as:  
\[\begin{cases}K_{I}(\theta)\\ K_{II}(\theta)\end{cases}=\sqrt{2\pi L}\mathbf{\Psi}_{L}^{(s)}(\theta)\mathbf{c^{(s)}} \tag{51}\]The use of the matrix order of singularity automatically accounts for special cases in material interfaces. This is achieved by its off-diagonal terms [50]. Consequently, the SBFEM does not pose any a priori assumption on the type of singularity, which greatly facilitates the simulation of crack propagation through heterogeneous media.  
Enhancing SIFs  
Since the SIFs are directly evaluated using singular stress modes, standard recovery techniques may be applied, in order to improve on the solution during post-processing. Two pertinent methods are the Superconvergent patch recovery (SPR) theory [218; 219; 220] and curve fitting by splines [221]. In the former, an improved estimation of the singular stresses is obtained by smoothing the singular stress modes by means of SPR theory. The main benefit originates in the availability of error estimators [217] and the theoretical underpinning of the method. The latter is highly pragmatic and empirically offers comparable accuracy at reduced computational cost. Differing from the SPR method, the singular stresses computed at the Gauss points are fitted using a spline.