## 2 Models and Methods
### Finite Element Formulation  
A two-dimensional in-plane dynamic rupture model of two blocks separated by a planar interface was studied numerically by using the finite element method [36; 37; 38]. The conservation of energy can be expressed as [39]:  
\[\frac{d}{dt}\!\!\int\limits_{\dot{A}}\!\!\left(\frac{1}{2}\rho\mathbf{v}\cdot \mathbf{v}+\rho U\right)\!\!dA\ =\ \left(\int\limits_{S}\mathbf{v}\cdot\mathbf{T}^{t}\!\!dS+\int\limits_{A} \mathbf{f}\cdot\mathbf{v}dA\right)-\left(-\int\limits_{S}\mathbf{v}\cdot \mathbf{T}^{t}\!\!dS\right)-\left(-\int\limits_{S}\mathbf{v}\cdot\mathbf{T}^{ t}\!\!dS\right)\equiv\dot{E}_{W}-\dot{E}_{F}-\dot{E}_{QB} \tag{1}\]  
where \(t\) is time, \(\rho\) is the mass density, \(\mathbf{v}\) is the velocity field vector, \(U\) is the internal energy in terms of unit mass, \(\mathbf{f}\) is the body force vector, \(\mathbf{T}^{t}\) is the surface distributed load, \(\mathbf{T}^{t\!\!\beta}\) is the solid infinite element radiation traction, \(\mathbf{T}^{t}\) is the frictional traction, \(\dot{E}_{W}\) is the rate of work done to the body by external forces, \(\dot{E}_{QB}\) is the energy that is dissipated by the damping effect of solid medium infinite elements, and \(\dot{E}_{F}\) is the rate of the energy that is dissipated by contact friction forces between the contact surfaces. Substituting the discrete expression of displacements into Equation (1), we can obtain the equations of motion for nodal displacements in finite element method,  
\[M\ddot{\mathbf{u}}(t)+C\dot{\mathbf{u}}(t)+K\mathbf{u}(t)\ =\ \mathbf{Q}(t) \tag{2}\]  
where M, C, K, and Q(t) denote the mass, damping, stiffness matrices, and the force vector at time t. \(\ddot{\mathbf{u}}(t)\), \(\dot{\mathbf{u}}(t)\) and \(\mathbf{u}(t)\) represent acceleration, velocity and displacement at time t, respectively. The force vector can be expressed as \(\mathbf{Q}(t)\ =\ \mathbf{F}_{t}(t)-\mathbf{F}_{f}(D(t),\dot{D}(t))\), where \(\mathbf{F}_{t}(t)\) and \(\mathbf{F}_{f}(D(t),\dot{D}(t))\) are the tectonic force and friction force vectors, respectively, D denotes the slip on the fault, and \(\dot{D}\) denotes the sliding velocity on the fault. We used the dynamic finite element method for discontinuous contact mechanics to simulate the slide on the fault in the calculation. Supposing the normal displacements on the fault meet continuous condition on both sides of the fault surfaces, we can obtain:  
\[\mathbf{u}^{\mathrm{n}}_{-}-\mathbf{u}^{\mathrm{n}}_{+}\ =\ 0 \tag{3}\]  
In addition, we suppose that stresses on the fault surface satisfy the following conditions [40]:  
\[\left\{ \begin{array}{ll}|\tau|<\mu\sigma_{n},&\mathrm{stick}\\ \tau\ =\ -\mu\sigma_{n},&\mathrm{slip}\end{array} \right. \tag{4}\]  
where \(\sigma_{\mathrm{n}}\) is the normal contact stress, \(\tau\) is the shear stress, and \(\mu\) is the coefficient of friction. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Description** & **Parameter** & **Value** \\ \hline Shear wave speed & \(\mathrm{V_{s}}\) (m/s) & 3333 \\ Dilatational wave speed & \(\mathrm{V_{P}}\) (m/s) & 5774 \\ Initial normal stress & \(\mathrm{\sigma_{0}}\) (MPa) & 50 \\ Initial shear stress & \(\mathrm{\tau_{0}}\) (MPa) & 28 \\ Density & \(\rho\) (kg/m\({}^{3}\)) & 2700 \\ Poisson’s ratio & \(v\) & 0.25 \\ Shear stress within nucleation zone & \(\mathrm{\tau_{nucl}}\) (MPa) & 31.7 \\ Barrier strength & \(\mathrm{\Delta\sigma_{n}/\sigma_{n}}\) & 1\%\(\sim\)16\% \\ Nucleation size & \(\mathrm{L_{nz}}\) (km) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters used in simulations. Based on the above Equations (2)-(4), a displacement solution is obtained by the finite element method. The time integration algorithm plays an important role in FEM calculations, especially in solving nonlinear problems. In order to guarantee calculation precision and enhance computing efficiency, we can use the central difference method to solve the Equations (2)-(4) (the explicit finite element method).