# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 3 Domain Reduction Method
To validate the 2-D DRM, we conducted a comparative analysis of the results obtained from the DRM model with the analytical solutions for inclined incident SV waves (case A) and the corresponding results from the 1-D site response analysis (case B). The material properties of the soil column utilized in the 1-D SRA are identical to those employed in the DRM model (see Figure 10d). The soil column and the domain of interest of the DRM model have the same height. The properties of the DRM model are outlined in Table 3. For all verification cases, a Ricker pulse is employed as the incident wave function, with parameters set to \(A_{Ricker}=10^{-4}\) m, \(f_{Ricker}=2\) Hz, \(t_{0}=0.8\) s. Here, \(A_{Ricker}\), \(f_{Ricker}\), and \(t_{0}\) are amplitude, dominant frequency, and time offset of the Ricker pulse. The total simulation time is 3 s, with an increment of \(\Delta t=0.001\) s. The time histories and Fourier amplitude spectra of the Ricker pulse are illustrated in Figure 11.  
Figure 11: Plots for the applied Ricker pulse include; (**a**) displacement, (**b**) Fourier amplitude, (**c**) velocity, and (**d**) acceleration. Figures 12 and 13 show the displacement histories for the selected points (refer to Figure 10d), acquired from the DRM, 1-D SRA, and the analytical solutions. The DRM results closely align with both the analytical solutions and 1-D SRA. Figure 12: Displacement histories for case A; (**a**) horizontal displacement at P1, (**b**) vertical displacement at P1, (**c**) horizontal displacement at P2, (**d**) vertical displacement at P2, (**e**) horizontal displacement at P3, and (**f**) vertical displacement at P3. After successfully implementing and validating the PMDL and DRM individually, the integrated PMDL-DRM system was rigorously tested. The testing process involved three distinct models to evaluate the accuracy and reliability of this integrated system:  
1. The Extended Domain Model: this model (illustrated in Figure 14a) represents a detailed and comprehensive approach, serving as a reference or benchmark for comparison. 2. The PMDL-DRM Combined Model: depicted in Figure 14b, this model incorporates both the PMDL and DRM methodologies, blending their strengths to achieve a more refined simulation. 3. The Winkler Model: shown in Figure 14c, this simpler model is often used for foundational analysis and offers a different perspective on soil-structure interaction. The primary focus of the comparison was on the roof displacement of the frame structure, particularly the horizontal displacement at a specific point labeled as \"A\" in Figure 14a-c. This comparison was crucial in determining whether the PMDL-DRM system could accurately replicate the behavior of the structure under seismic loading. For the purpose of this verification, it was assumed that both the soil and the frame structure behaved elastically. The soil properties across all three models were kept consistent to ensure a fair comparison, with the specific values provided in Table 3. Additionally, the Young Modulus of material of the frame structure is \(32\times 10^{6}\) kPa and section properties of the frame structure are detailed in Figure 14a-c.  
Figure 13: Horizontal displacement histories for case B at (**a**) P1, (**b**) P2, and (**c**) P3. Figure 14: (**a**) Extended domain model, (**b**) PMDL-DRM combined model, (**c**) Winkler model, (**d**) acceleration time-history of the Kobe earthquake, and (**e**) comparison of the roof displacements of three distinct models. To simulate real-world conditions, a historical earthquake record--the acceleration time-history of the Kobe earthquake--was used as the input motion, as illustrated in Figure 14d. This record provided a realistic and challenging test for the models. The results of the analysis are depicted in Figure 14e. The roof displacements calculated by the PMDL-DRM system were found to be in excellent agreement with those obtained from both the extended model and the Winkler model.