# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 1 Introduction
[PERSON] and [PERSON] [23] applied complex coordinate stretching to modify the material matrix, while [PERSON] et al. [24] and [PERSON] et al. [25] used it to alter the strain-displacement matrix. The research on ABCs has resulted in the creation of a new class of continuous ABCs, designated as PMDLs. These PMDLs serve to bridge the gap between PML, and rational ABCs as reported by [26; 27; 28], among others. In essence, PMDLs represent a specifically discretized version of PML, with the consequence that they are equivalent to rational ABCs. This, in particular, is achieved by employing linear finite elements with midpoint  
Figure 1: (**a**) Physical problem; (**b**) Soil domain with special boundary condition. integration for discretization [29; 30]. It has been demonstrated that, when approximating the Dirichlet-to-Neumann (DtN) map, the error from midpoint integration cancels out the discretization error, resulting in a perfect match even after exterior discretization. This phenomenon has led to the designation of these discrete layers as \"perfectly matched discrete layers,\" or PMDLs. The sole source of error in the DtN map is the truncation of PMDLs, which diminishes exponentially with the increase in the number of layers. Initially, PMDLs were developed with the objective of efficiently implementing rational ABCs [27], and they have been demonstrated to be mathematically equivalent to rational ABCs [27]. Given their close connection to both PML and rational ABCs, PMDLs can be regarded as a unification of the two, combining the respective advantages of flexibility and efficiency. Various studies have validated the effectiveness of PMDLs for different situations e.g., static analysis, and scalar and harmonic wave propagation (see, among others, [26; 30; 31]). [PERSON] and [PERSON] [32] introduced a modified version of PMDLs for SSI analysis in the time-domain, whereas [PERSON] et al. employed PMDLs in layered media for nonlinear SSI analysis in the time-domain [33]. [PERSON] [34] proposed a numerical method incorporating PMDLs for layered poroelastic soil and further developed three-dimensional PMDL formulations as well [35]. Building on prior work, [PERSON] and [PERSON] applied the Reciprocal Absorbing Boundary Condition (RABC) in conjunction with PMDLs to facilitate transient analysis of in-plane wave motion within a layered half-space [36]. [PERSON] et al. applied PMDLs to the analysis of concrete gravity dams [37]. OpenSees, unlike commercial finite element software, is available for free, which boosts its popularity. Whereas several successfully implemented and widely used models for superstructures (e.g., reinforced concrete, steel, masonry, timber) are available in the OpenSees library, the same cannot be said for soil-structure interactions, where the sheer size of the domain and complexity of the problems render the use of currently available models/methods inefficient in terms of the analysis time and computational resources required. As models used to analyze soil-structure interactions become more complex, analysis times increase significantly. A pertinent example is that of performing seismic fragility analysis, which requires a very high number of simulations to account for the inherent uncertainties and variations in both earthquake events and structural responses. Therefore, the integration of PMDL and Domain Reduction Method (DRM) methods enhances OpenSees as a powerful tool for seismic analysis and in obtaining better results in less time compared to other boundary conditions discussed in the article. Hence, the addition of PMDLs and DRM to OpenSees facilitates earthquake engineering research by providing researchers with an open-source tool to analyze soil-structure interactions. In this study, we utilize the PMDL formulation proposed by [PERSON] and [PERSON] [32] by developing a custom element in OpenSees (3.5) [38] using the Extended Newmark-\(\beta\) implicit time integration scheme for simulating wave propagation in two-dimensional (2-D) half-spaces.