## 3 Free-Field One-Dimensional Time Domain Algorithm Based on the Mass Method  
The incident wave used in reference [23] was an upward wave, while the response of the bottom node was s also affected by the downward wave. When a deep well borehole ground motion record is known or for a given input to a vibration table, the ground motion input is the total motion time history. For the problem of simulating the given input, the mass method can be used to solve the motion equation of the vertical soil node.  
The finite element discretization model of horizontal layered soil layer is shown in Figure 3. In the finite element method of the concentrated mass method, the restoring force and damping force of any one node are only related to the movement of the adjacent nodes [24]. If the displacement of node (\(m\), \(n\)) at moment \(t=p\Delta t\) is denoted as \(\mathbf{u}_{m,n}^{p}=\left(\begin{array}{cc}u_{m,n}^{px}&u_{m,n}^{py}\\ \end{array}\right)^{T}\), the equation of motion of the node (\(m\), \(n\)) at t moment can be expressed as shown below.  
\[\mathbf{M}_{m,n}\tilde{u}_{m,n}^{p}+\sum_{i=m-1}^{n+1}\sum_{j=n-1}^{n+1}\left(\bm {C}_{i,i}\mathbf{i}_{i,j}^{p}+\mathbf{K}_{i,i}\mathbf{u}_{i,j}^{p}\right)=\mathbf{P}(t) \tag{3}\]  
where \(\mathbf{M}_{m,n}\) is the node mass matrix; \(\mathbf{C}_{i,j}\) is the damping coefficient matrix, with the subscript indicating the damping force of unit velocity of j to (i); \(\mathbf{K}_{i,j}\) is the stiffness matrix coefficient, with the subscript indicating the restoring force of unit displacement of j to (i); and \(\mathbf{P}(t)\) is the external force received by the node. When the input excitation is specified for the bottom, the boundary node is treated by the mass method [25]; that is, the boundary node \(\mathbf{M}_{m,N}\) is given a large mass (which may be \(10^{5}\) of the total mass), and the corresponding seismic excitation equivalent load is \(\mathbf{P}(t)=\mathbf{M}_{m,N}\tilde{u}_{g}(t)\). This will generate additional damping force for large mass points when considering Rayleigh damping,  
Figure 2: Finite element model of soil-structure dynamic interactions.  
the wave velocity of P wave or SV wave is \(c\), [PERSON]'s law [28] dictates that the horizontal apparent wave velocity \(c_{x}\), propagation velocity \(c\), and incident angle \(\theta\) of the incident wave are related as follows:  
\[c_{x}=\frac{c}{\sin\theta} \tag{8}\]  
Figure 5 is the schematic diagram of the displacement expansion solution for the free wave field. After calculating the displacement of a column of nodes, the expansion can be solved according to the propagation law of the wave in the medium:  
\[u(x,y,t)=u(0,y,t-x/c_{x}) \tag{9}\]  
When a P or SV wave is incident vertically, the horizontal apparent wave velocity is infinite (Equation (8)). According to Equation (9), the displacement of all boundary nodes \(\mathbf{u}_{B}\) and that of the nodes connected to the boundary nodes directly \(\mathbf{u}_{I}\) can be obtained. By substituting the solution into Equation (2), we can obtain the equivalent load \(\mathbf{F}_{B}\) on the boundary nodes. The above is the solution of the free-field two-dimensional plane model. For the three-dimensional model, the expansion solution can be obtained by using consistent displacement in the outward plane direction.