**3. Input of Seismic Waves in Numerical Simulation**

*3.1. Establishment of the Artificial Boundary and the Input Method*

In a dynamic analysis of the numerical simulation, the study area is usually truncated from an infinite region with an artificial boundary used to absorb the scattered waves. In this study, a viscous-spring artificial boundary was adopted [54,55]. The elastic springs and dampers were established to implement the viscous-spring artificial boundary on the boundary nodes, as shown in Figure 2. The coefficients of the elastic spring (*K*) and damper (*C*) are defined as follows.

$$K\_{\rm N} = \frac{1}{1+A} \frac{\lambda + 2G}{R} \cdot A\_{I\prime} \cdot \mathbb{C}\_{\rm N} = B \rho c\_{\rm P} \cdot A\_{I} \tag{1}$$

$$K\_{\rm T} = \frac{1}{1+A} \frac{G}{R} \cdot A\_{l\prime} \cdot \mathbb{C}\_{\rm T} = B\rho c\_{\rm s} \cdot A\_{l} \tag{2}$$

where *A* and *B* are the correction coefficients; the good values of the coefficients are 0.8 and 1.1, respectively [56]; *ρ* is the mass density; *R* is the distance from the wave source to the boundary; *c*<sup>p</sup> and *c*<sup>s</sup> are the velocities of the compression wave and shear wave in the medium, respectively; *G* is the shear modulus; and subscripts T and N indicate the tangential and normal directions, respectively. *Al* represents the influence area at each node, for example, at node *l*, *Al* = (*A*<sup>1</sup> + *A*2)/2, as depicted in Figure 2.

**Figure 2.** Layout of viscous-spring artificial boundary on FEM model.

The motion equation of the lumped mass in the FEM wavefield at the artificial boundary is expressed as follows:

$$m\ddot{\mathbf{u}} + c\dot{\mathbf{u}} + k\mathbf{u} = A\sigma \tag{3}$$

In case of the distribution of the wavefield on the artificial boundary, the displacement and stress on the artificial boundary can be divided into free-field motions (denoted by superscripts *f*) and scattered-field motions denoted by superscripts *s*), which are illustrated as follows:

$$\mathbf{u} = \mathbf{u}^f + \mathbf{u}^s \boldsymbol{\sigma} = \mathbf{o}^f + \mathbf{o}^s \tag{4}$$

The equation of stress (specifically for the scattered-field motions) on the viscousspring artificial boundary is given as:

$$
\sigma^s = -K\mathbf{u}^s - \mathbf{C}\dot{\mathbf{u}}^s \tag{5}
$$

Substituting Equations (4) and (5) into Equation (3), the motion equation on the artificial boundary is expressed as follows:

$$m\ddot{\mathbf{u}} + (c + AC)\dot{\mathbf{u}} + (k + AK)\mathbf{u} = A\sigma + K\mathbf{u}^f + C\dot{\mathbf{u}}^f \tag{6}$$

Equation (6) can be considered seismic input and non-radiation, and the right side of the equation represents the equivalent nodal forces. When the two directions of the viscousspring artificial boundary (normal direction and tangential direction) are considered, the equivalent nodal force can be expressed as follows:

$$\mathbf{f}\_{li} = \mathbf{K}\_{li}\mathbf{u}\_{li}^{f} + \mathbf{C}\_{li}\dot{\mathbf{u}}\_{li}^{f} + A\_{l}\mathbf{w}\_{li}^{f} \tag{7}$$

where subscript *i* denotes the direction and subscript *l* denotes the node. *Kli* and *Cli* are the coefficients of the elastic spring and damper, which can be calculated using Equations (1) and (2), respectively; σ*<sup>f</sup> li*, **<sup>u</sup>***<sup>f</sup> li* and **. u***f li* represent the tensors of the stress, displacement and velocity, respectively.
