# A Partitioned Rigid-Element and Interface-Element Method for Rock-Slope-Stability Analysis
## 2 Theory
### Equations of PRE-IE  
In order to make the force and displacement of the rigid element and interface element transfer correctly on the contact surface, the rigid element is partitioned so that the rigid element and interface element have corresponding nodes on the contact surface (as shown in Figure 1c). It is also necessary to make the following assumptions about the distribution modes of contact force and nodal displacement so that the force and displacement of the rigid element and the interface element can be coupled on the contact surface:  
\[\mathbf{a}^{\varepsilon}=\mathbf{W}\mathbf{\gamma}^{\varepsilon}\qquad\text{on }S_{i} \tag{17}\]  
\[\mathbf{F}=\mathbf{N}_{p}\mathbf{f}^{\varepsilon}\qquad\text{on }S_{i} \tag{18}\]  
where \(\mathbf{N}_{p}\) is the interpolation function matrix. By setting different interpolation functions, the contact force can be interpolated to obtain the linear distribution, quadratic distribution, or other forms of distribution surface force. In the following numerical examples, the surface forces are assumed to be linearly distributed. \(\mathbf{f}^{\varepsilon}\) is the contact force array of contact node pairs on the interface element.  
By combining Equations (5) and (16), the potential energy equation of PRE-IE can be obtained. By taking Equations (17) and (18) as constraint conditions, using the Lagrange multiplier method, the modified functional equation as shown below can be obtained :  
\[\begin{split}\Pi_{P}^{\varepsilon}&=\int_{S_{\
u}} \mathbf{\gamma}^{\varepsilon^{T}}\mathbf{W}^{T}\mathrm{d}S+\int_{S_{i}}\mathbf{\gamma}^{ \varepsilon^{T}}\mathbf{W}^{T}\mathbf{\mathrm{F}}\mathrm{d}S+\int_{V}\mathbf{\gamma}^{ \varepsilon^{T}}\mathbf{W}^{T}\mathbf{f}_{g}\mathrm{d}V\\ &+\frac{1}{2}\int_{\Omega}\frac{1}{\mathbf{t}^{2}}\mathbf{a}^{ \varepsilon^{T}}\mathbf{B}^{T}\mathbf{D}\mathbf{B}\mathbf{a}^{\varepsilon}\mathrm{d}V-\int_{S_ {i}}\mathbf{a}^{\varepsilon^{T}}\mathbf{N}^{T}\mathbf{\mathrm{F}}\mathrm{d}S\\ &+\int_{S_{i}}\lambda(\mathbf{a}^{\varepsilon}-\mathbf{W}\mathbf{\gamma}^{ \varepsilon})\mathrm{d}S+\int_{S_{i}}\mu\big{(}\mathbf{F}-\mathbf{N}_{p}\mathbf{f}^{ \varepsilon}\big{)}\mathrm{d}S\end{split} \tag{19}\]  
The total potential energy \(\Pi_{P}\) of a structure is the sum of the potential energy of each element:  
\[\Pi_{P}=\sum_{\varepsilon}\Pi_{P}^{\varepsilon} \tag{20}\]  
The above equation requires that the order of each matrix in the element is the same as that in the overall structure, so the following transformation matrix needs to be introduced:  
\[\mathbf{f}^{\varepsilon}=\mathbf{G}_{f}\mathbf{f} \tag{21}\]  
\[\mathbf{\gamma}^{\varepsilon}=\mathbf{G}_{\gamma}\mathbf{\gamma} \tag{22}\]  
where \(\mathbf{G}_{f}\) and \(\mathbf{G}_{\gamma}\) are the transformation matrixes, \(\mathbf{f}\) is the contact force array of overall structure, and \(\mathbf{\gamma}\) is the rigid displacement array of the overall structure.  
The governing equation of PRE-IE shown below can be obtained from Equation (20) by using the variational calculus:  
\[\left[\begin{array}{cc}K_{ff}&K_{f\gamma}\\ K_{\gamma f}&K_{\gamma\gamma}\end{array}\right]\left[\begin{array}{c}f\\ \gamma\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{P}\end{array}\right] \tag{23}\]  
where  
\[K_{ff}=-\sum_{\varepsilon}\int_{S_{i}}\mathbf{G}_{\gamma}{}^{T}\mathbf{W}^{T} \mathbf{N}^{T}\mathbf{N}_{p}\mathbf{G}_{f}\mathrm{d}S \tag{24}\]  
\[K_{f\gamma}=\sum_{\varepsilon}\int_{\Omega}\frac{1}{t^{2}}\mathbf{G}_{\gamma}{ }^{T}\mathbf{W}^{T}\mathbf{B}^{T}\mathbf{D}B\mathbf{W}\mathbf{G}_{\gamma} \mathrm{d}V \tag{25}\]  
\[K_{\gamma f}=-\sum_{\varepsilon}\int_{S_{i}}\mathbf{G}_{\gamma}{}^{T}\mathbf{ W}^{T}\mathbf{N}_{p}\mathbf{G}_{f}\mathrm{d}S \tag{26}\]  
\[K_{\gamma\gamma}=\mathbf{0} \tag{27}\]  
\[\mathbf{P}=\sum_{\varepsilon}\biggl{(}\int_{S_{\sigma}}\mathbf{G}_{\gamma}^{ T}\mathbf{W}^{T}\mathrm{d}S+\int_{V}\mathbf{G}_{\gamma}^{T}\mathbf{W}^{T} \mathbf{f}_{g}\mathrm{d}V\biggr{)} \tag{28}\]  
The contact force \(f\) and rigid displacement \(\gamma\) can be obtained by solving the Equation (23).