## 4 The Scaled Boundary Finite Element Method (SBFEM)
### Principles of the Scaled Boundary Finite Element Method  
The characteristic trait of SBFEM, setting it apart from other numerical methods, and enabling an elegant computation of the gSIFs, is the introduction of a scaling center. Each polygonal SBFEM element, referred to as a subdomain, may only contain one scaling center, from which the whole boundary must be visible. By consequence, a new reference system is introduced with a radial coordinate \(\xi\) and a local tangential coordinate \(\eta\) (Figure 2a). These resembles a polar reference frame, and are termed the _scaled boundary coordinates_. The theoretical basis of the SBFEM is summarized in [186] and more recently and extensively in [212]. In this work, only the fundamental features are discussed. A thorough and more extensive treatment of the latest SBFEM-advancements in the context of LEFM can be found in the recent review paper by Song et al. [213]. Figure 2: Domain discretisation, scaling center \(O\) and introduction of scaled boundary coordinates. (**a**) Polygon domain with scaled boundary coordinates; (**b**) Transformation of singular stress field around crack tip to polar coordinates. Considering 2D bounded domains, the radial coordinate ranges from \(0<\xi<1\), initiating at the scaling center and ending on the boundary. This component is kept analytical throughout the analysis, thereby reducing the dimensionality of the problem by one. Hence, only the boundary of the subdomain requires discretization, in the finite element sense, into independent line elements. For each line element, a separate local coordinate \(\eta\) is introduced with \(-1<\eta<1\). The mapping between Cartesian \((x,y)\) and scaled boundary coordinates \((x(\xi,\eta),y(\xi,\eta))\) is achieved by scaling points \((x_{b},y_{b})\) on the boundary: For a given set of nodal coordinates \(\mathbf{x_{b}},\mathbf{y_{b}}\) and conventional finite element shape functions \(\mathbf{N(\eta)}\) the below mapping results in:  
\[x(\xi,\eta) =\xi x_{b}(\eta)=\xi\mathbf{N(\eta)}\mathbf{x_{b}} \tag{28}\] \[y(\xi,\eta) =\xi y_{b}(\eta)=\xi\mathbf{N(\eta)}\mathbf{y_{b}} \tag{29}\]  
which employs \(\xi\) as a scalar. Similarly the displacements contain an analytical and an interpolatory component:  
\[\mathbf{u(\xi,\eta)}=\mathbf{N^{u}(\eta)u(\xi)}=(\mathbf{N_{1}(\eta)I,...,N_{n}(\eta)I})\bm {u(\xi)} \tag{30}\]  
The subscript \(n\), denotes the degrees of freedom (DOF) present in the line element. In 2D, \(\mathbf{I}\) is a \(2\times 2\) identity matrix and \(\mathbf{u(\xi)}\) are nodal displacement functions along a line connecting the scaling center and the boundary. Consequently, the displacements on the boundary are synonymous with \(\mathbf{u}=\mathbf{u(\xi=1)}\). The expression of the stress follows as [214]:  
\[\mathbf{\sigma(\xi,\eta)}=\mathbf{D(B^{1}(\eta)u(\xi),_{\xi}}+\mathbf{B^{2}(\eta)u(\xi)/ \xi)} \tag{31}\]  
\(\mathbf{D}\) represents the constitutive matrix. \(\mathbf{B^{1}(\eta)}\) and \(\mathbf{B^{2}(\eta)}\) together describe the strain-displacement relation [212]. Once the governing differential equation is rewritten in scaled boundary coordinates, standard techniques such the Galerkin's weighted residual method [186], the principle of virtual work [187] or the Hamiltonian principle [215] may be applied in the circumferential direction \(\eta\) giving rise to the two governing equations of SBFEM, i.e., Equations (32) and (33). \[\mathbf{E^{0}\xi^{2}u(\xi),_{\xi\xi}+(E^{0}-E^{1}+E^{1^{T}})\xi u(\xi),_{\xi}-E^{2}u(\xi)}=0 \tag{32}\] \[\mathbf{P}=\mathbf{E^{0}\xi u,_{\xi}+E^{1^{T}}u}\text{ or in modal form }\mathbf{q}=\mathbf{E^{0}\xi u(\xi),_{\xi}+E^{1^{T}}u(\xi)} \tag{33}\]  
Equation (32), termed the scaled boundary finite element atomic displacement describes the behavior within the domain. Equation (33) expresses the behavior on the boundary, where \(\mathbf{P}\) comprises the vector of nodal forces.