# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 3 Domain Reduction Method
\begin{bmatrix}0\\ -M^{O\!2+}_{\text{be}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
The subscripts \(e,\ i,\ b\) denote the nodes between the ABCs layer and the DRM layer, nodes within the domain of interest, and nodes between the DRM layer and the domain of interest, respectively, and where \(u^{0},\ i^{0},\ and\ \bar{u}^{0}\) represent the free-field displacement, velocity, and acceleration, respectively. To utilize the coupled PMDL-DRM method in OpenSees, the main steps are as follows:  
1. Define the physical domain geometry, identify the boundaries for PMDL and DRM application, and generate the mesh.
2. Extract the nodes and elements corresponding to the DRM layer from the mesh.
3. Use the flowchart given in Appendix A (see Figure A1) to compute the mass, stiffness, and damping matrices (\(M^{\Omega+},C^{\Omega+},\ K^{\Omega+}\)) for the DRM layer.
4. Calculate the free-field responses (displacement, velocity, and acceleration) for the nodes in the DRM layer using 1D SRA or analytical methods.
5. Determine the DRM layer forces (\(P^{eff}\)) using Equation (29).
6. Create an OpenSees model and input the calculated DRM layer forces into it.  
The flowchart illustrating the sequential steps to implement and use the coupled PMDL-DRM approach in OpenSees can be found in Appendix A (see Figure A2).  
Calculating the free-field response for inclined incident SV waves (refer to Figure 10c) using 1D SRA is extremely difficult, as analytical solutions derived by [47] provided in (30)-(32) exist only for a homogeneous half-space in the time-domain. In Equations (30)-(32), \(U_{s}^{i},\ U_{s}^{r},\ U_{s}^{r}\) and \(U_{p}^{r}\) are the amplitudes of the incident SV wave, reflected SV wave, and reflected P wave, respectively. \(V_{s}\) and \(V_{p}\) denote the velocities of S-waves and P-waves. The variable \(f=f(t)\) refers to a time-dependent source-time function. \(u_{x^{\prime}}\) and \(u_{z}\) correspond to the displacement components for inclined incident SV waves. \(\theta_{s}\) and \(\theta_{p}\) are the angles of SV wave incidence and reflected P-wave, respectively.  
Figure 10: (**a**) Modelling of a half-space domain using ABCs and DRM; (**b**) equivalent 1-D SRA for estimation of free-field response for DRM; (**c**) schematic representation of SV wave propagation in a 2D homogeneous half-space; (**d**) numerical model constructed for DRM verification problem.  
\[\begin{bmatrix}u_{xt}(xt,z)\\ u_{z}(xt,z)\end{bmatrix}=U_{s}^{i}\begin{bmatrix}cos\theta_{s}\\ sin\theta_{s}\end{bmatrix}f\left(-\frac{\gamma^{\prime}}{V_{s}}\text{sin}\theta_ {s}+\frac{z}{V_{s}}\text{cos}\theta_{s}+t\right)+U_{s}^{r}\begin{bmatrix}-cos \theta_{s}\\ sin\theta_{s}\end{bmatrix}f\left(-\frac{\gamma^{\prime}}{V_{s}}\text{sin} \theta_{s}-\frac{z}{V_{s}}\text{cos}\theta_{s}+t\right)\\ +U_{p}^{r}\begin{bmatrix}sin\theta_{p}\\ cos\theta_{p}\end{bmatrix}f\left(-\frac{\gamma^{\prime}}{V_{p}}\text{sin} \theta_{p}-\frac{z}{V_{p}}\text{cos}\theta_{p}+t\right)\\ \\ U_{s}^{r}=\frac{\text{sin}(2\theta_{s})\text{sin}(2\theta_{p})-\left(\frac{V_{ p}}{V_{s}}\right)^{2}\text{cos}^{2}(2\theta_{s})}{\text{sin}(2\theta_{p})+ \left(\frac{V_{p}}{V_{s}}\right)^{2}\text{cos}^{2}(2\theta_{s})}U_{s}^{i}\\ U_{p}^{r}=\frac{-2\left(\frac{V_{p}}{V_{s}}\right)^{2}\text{sin}(2\theta_{s}) \text{cos}(2\theta_{s})V_{s}}{\text{sin}(2\theta_{s})\text{sin}(2\theta_{p})+ \left(\frac{V_{p}}{V_{s}}\right)^{2}\text{cos}^{2}(2\theta_{s})V_{p}}U_{s}^{i} \end{bmatrix}  \tag{30}\]  
\[U_{s}^{r}=\frac{\text{sin}(2\theta_{s})\text{sin}(2\theta_{p})-\left(\frac{V_ {p}}{V_{s}}\right)^{2}\text{cos}^{2}(2\theta_{s})}{\text{sin}(2\theta_{s}) \text{sin}(2\theta_{p})+\left(\frac{V_{p}}{V_{s}}\right)^{2}\text{cos}^{2}(2 \theta_{s})V_{p}}U_{s}^{i} \tag{31}\]  
\[U_{p}^{r}=\frac{-2\left(\frac{V_{p}}{V_{s}}\right)^{2}\text{sin}(2\theta_{s}) \text{cos}(2\theta_{s})V_{s}}{\text{sin}(2\theta_{s})\text{sin}(2\theta_{p}) +\left(\frac{V_{p}}{V_{s}}\right)^{2}\text{cos}^{2}(2\theta_{s})V_{p}}U_{s}^{i} \tag{32}\]  
In this investigation, the 2-D DRM was implemented in OpenSees.