## 4 The Scaled Boundary Finite Element Method (SBFEM)
### Principles of the Scaled Boundary Finite Element Method
The coefficient matrices \(\mathbf{E^{0},E^{1},E^{2}}\) are conceptually analogous to a subdomain stiffness matrix in the FEM: They are calculated for each element individually and then assembled. A general solution to the scaled boundary finite element equation in displacements, i.e., the homogeneous set of Euler-Cauchy differential equations in \(\xi\), is sought in its simplest form as a power series:  
\[\mathbf{u(\xi)}=c_{1}\xi^{-\lambda_{1}}\mathbf{\phi_{1}}+...+c_{n}\xi^{-\lambda_{n}} \mathbf{\phi_{n}}=\mathbf{\phi_{\mathrm{o}}}\xi^{-\lambda}\mathbf{c} \tag{34}\]  
The calculation of the eigenvalues, \(\lambda_{i}\), and eigen-vectors, \(\phi_{i}\), by means of eigen-decomposition can result in numerical errors, when near parallel eigen-vector pairs are present [216]. To alleviate this, the block-diagonal Schur decomposition may be adopted [217]. The displacement solution is obtained as a superposition (Figure 3) of the modes, with associated scaling values, and constrained by integration constants \(c_{i}\), as obtained from the imposed boundary conditions. \[\mathbf{u(\xi)}=\mathbf{\
abla^{(u)}\xi^{-S}c}=\sum_{i=1}^{n}\mathbf{\
abla_{i}^{(u)}\xi^{ -S_{i}}c_{i}} \tag{35}\]The transformation matrix \(\mathbf{\
abla}\) and block diagonal real Schur form \(\mathbf{S}\) are derived from recasting the system of first order differential equations (Equations (32) and (33)) as:  
\[\xi\left\{ \begin{aligned} \mathbf{u}(\mathbf{\xi})\\ \mathbf{q}(\mathbf{\xi})\end{aligned} \right\}_{\xi}=-\mathbf{Z}\left\{ \begin{aligned} \mathbf{u}(\mathbf{\xi})\\ \mathbf{q}(\mathbf{\xi})\end{aligned} \right\} \tag{36}\]  
with the Hamiltonian coefficient matrix \(\mathbf{Z}\) defined by  
\[\mathbf{Z}= \begin{bmatrix}\mathbf{E}^{0^{-1}}\mathbf{E}^{1^{T}}&-\mathbf{E}^{0^{-1}} \\ -\mathbf{E}^{2}+\mathbf{E}^{1}\mathbf{E}^{0^{-1}}\mathbf{E}^{1^{T}}&-\mathbf{E}^{1}\mathbf{E}^{0^{-1} }\end{bmatrix}  \tag{37}\]  
so that Equation (36) is decoupled by the block-diagonal Schur decomposition. \[\mathbf{Z}\mathbf{\
abla}=\mathbf{\
abla}S \tag{38}\]  
The columns of the transformation matrix contain the modes, whereas the diagonal blocks of the real Schur form contain the corresponding eigenvalues. However, Equation (36) results in doubling the amount of DOFs present in the solution, which can be shown to contain a bounded response (\(0<\xi<1\) and negative eigenvalues) and an unbounded response (\(1<\xi<\infty\) and positive eigenvalues). \(\mathbf{S}\) and \(\mathbf{\
abla}\) are sorted in ascending order and partitioned accordingly. \[\mathbf{S}=diag(\mathbf{S}_{neg},\mathbf{S}_{pos}) \tag{39}\]  
\[\mathbf{\
abla}= \begin{bmatrix}\mathbf{\Psi}_{neg}^{(\mathbf{u})}&\mathbf{\Psi}_{pos}^{(\mathbf{ u})}\\ \mathbf{\Psi}_{neg}^{(\mathbf{q})}&\mathbf{\Psi}_{pos}^{(\mathbf{q})}\end{bmatrix}  \tag{40}\]  
By expressing the nodal forces on the boundary with enforced integration constants (Equation (35) in Equation (33)), an expression for the stiffness matrix of the subdomain is obtained and a displacement solution is calculated analogous to FEM:  
\[\mathbf{K}_{bounded}=\mathbf{\Psi}_{pos}^{(\mathbf{q})}\mathbf{\Psi}_{neg}^{(\mathbf{u})-1} \tag{41}\]  
Finally, the stresses are obtained by substituting Equation (35) into Equation (31):  
\[\mathbf{\sigma}(\mathbf{\xi},\mathbf{\eta})=\mathbf{\
abla}_{\sigma}(\mathbf{\eta})\xi^{-\mathbf{S}_{ neg}-\mathbf{I}}\mathbf{c} \tag{42}\]  
where \([\mathbf{\Psi}_{\sigma i}(\mathbf{\eta})]\) is the stress mode of the corresponding displacement mode \([\mathbf{\Psi}_{i}^{(u)}]\). \[\mathbf{\
abla}_{\sigma}(\mathbf{\eta})=\mathbf{D}(-\mathbf{B}^{1}(\mathbf{\eta})\mathbf{\Psi}_{neg}^ {(\mathbf{u})}\mathbf{S}_{neg}+\mathbf{B}^{2}(\mathbf{\eta})\mathbf{\Psi}_{neg}^{(\mathbf{u})}) \tag{43}\]  
Figure 3: Graphical representation of modes. In black the original domain with linear elements and in gray the modes with corresponding values. The calculation of the stress on the domain boundary (\(\xi=1\)) does not require the evaluation of the matrix exponential \(\xi^{-S-I}\).