## 2 Discontinuity Layout Optimization  
The underlying concept of the DLO procedure is based on the _upper bound_ theorem [9]. Instead of \"hand calculations\", whereas a clearly defined collapse mechanism is chosen and analyzed, the DLO uses a large number of possible collapse mechanisms and searches for the mechanism with the lowest corresponding load. The collapse mechanisms are defined by dividing the geometry in a sufficient number of nodes. Each node is then connected to each other node by a discontinuity. A linear programming algorithm, including constraints, is used to find the collapse mechanism with the lowest resistance. The DLO concept is illustrated in Figure 2a-c, and the LP formulation can be seen from Equation (1). \[ \begin{array}{l}\min\lambda\cdot\ \underbrace{\mathbf{f}_{\mathrm{f}}^{ \mathrm{T}}\mathbf{d}}_{\mathrm{W_{ex}}=1}\ =\ \underbrace{\mathbf{g}^{\mathrm{T}}\mathbf{p}}_{\mathrm{W_{in}}}\ \text{such that}\ \left\{\begin{array}{l} \ \mathbf{B}\mathbf{d}=0\\ \mathbf{Np}-\mathbf{d}=0\\ \mathbf{f}_{\mathrm{L}}^{\mathrm{T}}\mathbf{d}=1\\ \mathbf{p}\geq 0\end{array}\right.\\ \end{array}  \tag{1}\]  
The global vectors \(\mathbf{d}\) and \(\mathbf{p}\) are the unknowns in the LP formulation, whereas \(\mathbf{d}\) contains the relative shear and normal displacement jumps (\(\mathrm{s_{i}}\) and \(\mathrm{n_{i}}\)) of each local discontinuities \(\mathbf{d_{i}}\) (compare Figure 2d), and \(\mathbf{p}\) is a vector containing the corresponding local plastic multipliers \(\mathbf{p_{i}}\). \(\mathbf{B}\) is a global compatibility matrix to ensure that at each node the sum of the shear and the normal displacement jumps equals zero (\(\mathbf{B}\mathbf{d}=0\)). The local compatibility matrix \(\mathbf{B_{i}}\) can be formulated with the direction cosines of each discontinuity i with:  
\[\mathbf{B_{i}}\mathbf{d_{i}}=\left[ \begin{array}{ccc}\cos\varphi_{\mathrm{i }}&-\sin\varphi_{\mathrm{i}}\\ \sin\varphi_{\mathrm{i}}&\cos\varphi_{\mathrm{i}}\\ -\cos\varphi_{\mathrm{i}}&\sin\varphi_{\mathrm{i}}\\ -\sin\varphi_{\mathrm{i}}&-\cos\varphi_{\mathrm{i}}\\ \end{array} \right]\left( \begin{array}{c}\mathrm{s_{i}}\\ \mathrm{n_{i}}\\ \end{array} \right) \tag{2}\]  
\(\mathbf{N}\) is the global plastic flow matrix. Contrary to the formulation in [8], the local flow matrix \(\mathbf{N_{i}}\) and the local plastic multiplier \(\mathbf{p_{i}}\) allows shear and normal displacement jumps at each discontinuity:  
\[\mathbf{N_{i}}\mathbf{p_{i}}-\mathbf{d_{i}}=\left[ \begin{array}{ccc}1&-1&0&0 \\ 0&0&1&-1\\ \end{array} \right]\left( \begin{array}{c}\mathrm{p_{si}^{+}}\\ \mathrm{P_{si}^{+}}\\ \mathrm{p_{ni}^{+}}\\ \mathrm{p_{ni}^{-}}\\ \end{array} \right)-\left( \begin{array}{c}\mathrm{s_{i}}\\ \mathrm{n_{i}}\\ \end{array} \right)=\left( \begin{array}{c}0\\ 0\\ \end{array} \right)=0 \tag{3}\]  
In doing so, slip line mechanisms and yield zones are allowed to form at each discontinuity. By modifying \(\mathbf{N_{i}}\), it is also possible to allow only slip lines or yield zones alone, which is shown in the further sections. Furthermore, the negative direction of the normal displacement jump of each discontinuity can be set to zero, which allows the implementation of a compression-only contact. Figure 2: Discontinuity layout optimization (DLO) procedure: (**a**) Geometry, loading, and boundary conditions; (**b**) discretization with nodes and discontinuities; (**c**) obtained collapse mechanism; (**d**) definition of a single discontinuity i. To formulate the (local) internal work of each discontinuity, the local vector \(\mathbf{g}_{i}\) is used. Corresponding to \(\mathbf{p}_{i}\), the vector \(\mathbf{g}_{i}\) contains the shear strength \(\tau_{y}\), the yield strength \(\mathrm{f}_{y}\), and the area \(\mathrm{A}_{i}=\mathrm{L}_{i}\mathrm{t}\) (\(\mathrm{L}_{i}\) and \(\mathrm{t}\) are the length and thickness, respectively) of discontinuity i:  
\[\mathbf{g}_{i}^{T}=\left( \begin{array}{cccc}\tau_{y}\mathrm{A}_{i}&\tau_{y} \mathrm{A}_{i}&\mathrm{f}_{y}\mathrm{A}_{i}&\mathrm{f}_{y}\mathrm{A}_{i}\end{array}