# Implementation of PMDL and DRM in OpenSees for Soil-Structure Interaction Analysis
## 1 Introduction  
Infinite domains (unbounded domains) are common in various research fields, especially in wave propagation studies. One particularly complex application in this area is soil-structure interaction (SSI). To tackle this, the soil model is split into two domains: a bounded (interior) domain, and an unbounded (exterior) domain (see Figure 0(a)). Both the structure and the bounded soil can display nonlinear behavior, while the unbounded soil is usually assumed to be linear. Precise solutions are typically needed only within the interior domain. Therefore, the unbounded soil is replaced by special boundary conditions (see Figure 0(b)) designed to absorb energy entering the medium and, hence, prevent it from reflecting back into the interior. These boundary conditions are known as absorbing boundary conditions (ABCs). Different ABCs (e.g., viscous boundaries, transmitting boundaries, non-reflecting boundaries, infinite elements, perfectly matched layers (PMLs), perfectly matched discrete layers (PMDLs)) are used to mimic the exterior domain. Viscous, transmitting, and non-reflecting boundaries are utilized in earthquake engineering simulations to absorb seismic waves, preventing them from reflecting back into the model and artificially amplifying the outcomes. These boundaries are crucial for accurately simulating open or semi-infinite domains by reducing artificial reflections and providing more realistic results [1; 2; 3; 4]. [PERSON] [5] introduced the first local ABC, designed to absorb waves traveling in a specific direction, whereas [PERSON] and [PERSON] [6] developed a technique to analyze the propagation of Love waves through structures with non-horizontal layers. Infinite elements are a type of finite element utilized in numerical simulations to represent problems that extend indefinitely, such as those involving wave propagation or electromagnetic fields. These elements mimic infinite domains by incorporating absorbing boundary conditions, which reduce wave reflections and create a seamless transition from the finite computational area to the infinite region. This method accurately represents far-field behavior without needing a large computational domain, improving efficiency in simulations of unbounded or semi-infinite domains [7; 8; 9; 10; 11; 12]. A particular type of continuous ABC, designated as PML, entails the substitution of the exterior with an absorbing medium (PML region) that exhibits precise impedance matching with the interior [13]. This substitution is terminated at a specific distance. This perfect impedance matching prevents reflections at the interface, while attenuation in the PML region serves to minimize reflections caused by truncation. The popularity of PML can be attributed to its flexibility and generality as a continuous ABC. Nevertheless, recent research indicates that PML may not be as effective as rational ABCs [14; 15]. [PERSON] et al. introduced a framework for implementing a new boundary condition that combines a perfectly matching layer with infinite elements, designed for addressing unbounded elastic wave problems in the time domain [16]. [PERSON] and [PERSON] presented time-harmonic governing equations for PMLs applicable to anti-plane and plane-strain motions in both elastic and visco-elastic media [17]. [PERSON] and [PERSON] [18] developed two- and three-dimensional viscoelastic PMLs that account for Rayleigh damping effects, which were then implemented in ABAQUS [19]. [PERSON] combined PMLs with frequency dependent finite element formulation [20]. [PERSON] et al. also developed a PML-based toolbox in ABAQUS (2017) for SSI analysis [21]. The key point of PMLs is the use of complex coordinate stretching, which employs complex-valued functions to enhance wave absorption and minimize boundary reflections in numerical simulations of wave propagation. [PERSON] and [PERSON] developed a coordinate stretch formulation for PMLs and applied it solely to two-dimensional wave propagation problems in homogeneous media, specifically for SH and SV wave propagation [22].