## 3 Mixed Numerical Discretization  
For lower bound solutions, a statically admissible stress field of the jointed rock mass is assumed. Over the past few decades, numerical methods, such as the FEM [21; 23; 24], the rigid finite element method [26; 30], and the rigid block method [28; 31; 32], together with mathematical programming techniques, have been developed and applied to solve limit analysis problems. In general, the strategies of these methods differ in terms of the numerical discretization and the assumptions made about the equilibrium equations and yield conditions. For the dual mechanical characteristics of the continuity and discontinuity of the non-persistent jointed rock mass, discrete methods are not suitable for establishing the statically admissible stress fields. A mixed numerical discretization of the lower bound limit analysis is then required and developed, in which the rigid block elements are used to simulate the rock blocks, and the finite element method is used to simulate the rock bridges (see Figure 1).  
A rigid block element is shown in Figure 2. To establish the statically admissible stress field for the rock blocks, it is convenient to define the following coordinate systems: \((x,y)\) = the global coordinate system, and \((n_{k},s_{k})\) = the local coordinate system for the interface \(k\) between the two adjacent block elements \(i\) and \(j\). Assume that (1) the forces at the interface \(k\) between the two adjacent block elements involve a normal force \(N_{k}\) and a shear force \(V_{k}\), both applied at the center of the interface, and (2) the forces at the center of block element \(i\) involve an external force \(f_{xi}\) in the \(x\) direction and an external force \(f_{yi}\) in the \(y\) direction. The forces can be denoted in vector form as  
\[\text{Q}_{k}=\begin{bmatrix}N_{k}&V_{k}\end{bmatrix}^{T} \tag{1}\]  
\[\text{F}_{i}=\begin{bmatrix}f_{xi}&f_{yi}\end{bmatrix}^{T} \tag{2}\]  
where \(k=(1,\cdots,n_{k})\), \(n_{k}\) is the number of all interfaces in all rock block elements; \(i=(1,\cdots,n_{R})\), \(n_{R}\) is the number of all block elements.  
Figure 1: Mixed numerical discretization of non-persistent jointed rock mass: (a) a non-persistent jointed rock slope contains three rock bridges; (b) mixed numerical discretization for jointed rock slope.  
To apply the lower bound limit theorem, the statically admissible stress field for the rock bridges needs to be established. To this end, the linear triangular finite element method proposed by [PERSON] is used to model the stress field of the rock bridges [21], as shown in Figure 3. The variation in the stress throughout each element is linear, and each node is associated with three unknown stress variables \(\sigma_{x}\), \(\sigma_{y}\) and \(\tau_{xy}\), where the tensile stresses are assumed as positive. Each node only belongs to a particular element, and more than one node may share the same coordinates (as shown in Figure 3a), which differs from the usual form of the finite element method. Statically admissible stress discontinuities are present on the element interface between adjacent triangular finite elements. The element stress can be denoted in vector form as:  
\[\sigma^{i}=\begin{bmatrix}\sigma^{i}_{x1}&\sigma^{i}_{y1}&\tau^{i}_{xy1}& \sigma^{i}_{x2}&\sigma^{i}_{y2}&\tau^{i}_{xy2}&\sigma^{i}_{x3}&\sigma^{i}_{y3}& \tau^{i}_{xy3}\end{bmatrix}^{T} \tag{3}\]  
where \(i=(1,\cdots,n_{e})\), and \(n_{e}\) is the number of triangular finite elements.