# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 2 Perfectly Matched Discrete Layers
### Verification of PMDL  
This section provides a thorough evaluation of the verification and stability of the implemented PMDL. During the verification process, the performance of the PMDL is compared to that of the built-in infinite element in ABAQUS. According to the ABAQUS manual, the infinite elements utilize a damping and reduced stiffness matrix. However, they do not perfectly transmit energy out of the mesh, except when plane body waves strike the boundary perpendicularly in medium. A half-space scenario (Figure 5a) is analyzed, with the relevant model properties detailed in Table 3. The excitation is provided by a vertical point load (see Figure 5b), which is represented by a Ricker pulse (see Figure 6). The amplitude, dominant frequency, and total duration of the Ricker pulse are 100 kN, 5 Hz, and 3 s, respectively. Two configurations are employed for the analysis: (i) PMDL elements (see Figure 5b), and (ii) infinite elements (replacing PMDL with CINPE4 [19]). Furthermore, an extended domain with dimensions of 1500 m by 750 m is constructed for the purpose of establishing a reference solution. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Acceleration Scheme** & \(\bar{\kappa}\) & \(\bar{\beta}\) & \(\bar{\gamma}\) \\ \hline Average & \(1/12\) & \(1/4\) & \(1/2\) \\ Linear & \(1/24\) & \(1/6\) & \(1/2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time increment values for Extended Newmark-\(\beta\) method. outer surface of the established reference solution is fixed at the horizontal and vertical directions. Figure 7 illustrates the comparisons of displacements obtained using the PMDL, infinite elements, and the extended domain for selected points. For points P1 and P2, only vertical displacements are shown, while for points P3 and P4, both horizontal and vertical displacements are presented. As depicted, the PMDL boundary performs excellently across all selected points, whereas the infinite elements do not perform as well. The infinite elements fail to perfectly absorb outgoing waves, leading to reflected waves re-entering the domain of interest. This results in discrepancies between the results from the domain using infinite elements and those from the PMDL and extended domain. These discrepancies are particularly evident in the vertical displacement at point P3. Additionally, Figure 8 displays contour plots of normalized displacement magnitudes at various times (t = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0, 3.0 s) using the PMDL boundary, further demonstrating its effectiveness in absorbing all outgoing waves. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline V\({}_{\text{s}}\) & 400 & m/s \\ \(\rho\) & 2000 & kg/m\({}^{3}\) \\ \(\
u\) & 0.3 & - \\ m & 9 & - \\ n & 2 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model properties used for verification. Figure 6: (**a**) Input force history and (**b**) Fourier amplitude. Figure 7: Comparison of displacements for the selected points; (**a**) vertical displacement at P1, (**b**) vertical displacement at P2, (**c**) horizontal displacement at P3, (**d**) vertical displacement at P3, (**e**) horizontal displacement at P4, and (**f**) vertical displacement at P4. A simulation with a total duration of 10 s was performed to assess the long-term stability of the PMDL. Figure 9 shows the displacement histories for the four selected points, whose locations are indicated in Figure 5b. The results reveal that the entire domain falls silent after approximately t = 1 s, with no reflections occurring between the PMDL boundary and the domain of interest. Another method to demonstrate the efficiency of the PMDL is by comparing the total computational time. For the extended domain using the ABAQUS solver on 20 processors, the total computational time is approximately 250 min. For the domain using infinite elements with the ABAQUS solver on 20 processors, it takes about 12 min. Meanwhile, the PMDL domain using the OpenSees solver on a single processor requires around 35 min.