## 4 The Scaled Boundary Finite Element Method (SBFEM)  
### An Abridged Literature Review of Advancements in SBFEM Fracture Modeling  
The SBFEM belongs to the class of semi-analytical methods and is therefore related to the thin layer method [178], the Trefftz method [179], the BEM [34], Spectral elements [180] and the semi-analytical finite elements [181]. SBFEM's key feature lie in introduction of a scaling center, which has been pioneered in the context of different domains, such as the solution of electric field problems [182]. Dasgupta et al. [183] refined and tailored the approach, which they termed the \"cloning algorithm\", to solid mechanics of unbounded media. Wolf and Song [184] subsequently adopted a similar formulation, which they termed the \"consistent infinitesimal finite-element cell method\". They later developed a standardized derivation relying on use of a weighted residual method [185; 186], and first coined the term \"SBFEM\". Later work by Deeks and Wolf [187] enabled broader adoption of the SBFEM method by introducing a virtual work based formulation.  
Although much of the early research focused on the treatment of unbounded domains, it was soon discovered that SBFEM is more effective at modelling bounded domains [186], particularly in the context of LEFM. This is apparent, since the fracture parameters, e.g., SIFs, T-stress as well as the coefficients of higher order terms, can be directly extracted from the singular components of the stress field [188; 189]. The method is able to robustly transition between power and power-logarithmic singularities [189]. It has thus been applied for computing the order of singularity and SIFs in multi-material plates under both static and dynamic loading [190], for predicting the crack propagation direction at bi-material notches [191], and for determining the free-edge stresses about holes in laminated composites [192].  
Yang et al. [193] first modelled crack propagation via use of SBFEM and a few large sized subdomains, whose initial meshes were manually specified. This approach was extended to model nonlinear cohesive fracture in concrete [194; 195; 196; 197; 198], dynamic fracture [199; 200] and crack propagation in reinforced concrete [201]. Reaching the limits of the laborious meshing approach, fully automated modelling of crack propagation was achieved by repurposing newly proposed meshers [202] for polygonal elements [203]. Currently, the most widely adopted meshing procedure combines the use of a quadtree decomposition with polygon clipping, to accurately represent curved geometries [51] with coarser meshes. The advantage of adopting balanced quadtree meshes as a basis lies in the limited amount of possible element realizations, whose pre-computation greatly enhances computational efficiency. Having resolved most mesh related issues, SBFEM was most recently extended to treat functionally graded materials [204; 205] and non-local damage [206; 207].  
An interesting development pertains to fusion of scaled boundary principles with IGA (SBIGA), which is shown to provide lower error in displacement and energy norm per degree of freedom. The method ensures exact treatment of curved boundaries [208; 209], delivers additional refinement possibilities and the ability to adjust continuity as required. However, the computational costs increased as compared against the standard SBFEM due to the integration procedure associated with IGA [210] partially due to the NURBS basis forming a larger support for the calculation of element related quantities [211]. When contrasted to established methods (e.g., FEM, IGA), this draw-back is negated as only the boundary need be discretised.