## 1 Introduction  
The dynamic soil-foundation-structure interaction (SFSI) is a complex, multidisciplinary field that encompasses civil, structural, and earthquake engineering. Due to its interdisciplinary nature, particular attention must be given to SFSI problems. The interaction between the soil and the structure influences the overall dynamic response, and neglecting this interaction can lead to inaccurate predictions on structural behavior, especially in terms of stress and strain levels. Additionally, during an earthquake, energy is dissipated through both the soil and the structure as the ground shakes. This dissipation reduces the forces transmitted to the structure, particularly in soft soils that deform and absorb more energy. Dynamic SFSI analysis is essential for developing designs that are both safe and cost effective. Recent studies on the SFSI have advanced readers' understanding of how soil dynamics influence structural responses during earthquakes [1; 2; 3; 4; 5]. Various physical and numerical models have been used to address these issues. The key challenge in dynamic SFSI analysis is the representation of unbounded soil behavior at infinity. Earlier research developed simple physical models to represent unbounded soil [6]. Physical models have been categorized into two types: cone models and lumped-parameter models. These models are particularly useful because they allow for pure time-domain computation, avoiding the abstraction of frequency-domain transformation and enabling the consideration of nonlinear structural behavior. Time-domain analysis also facilitates the assessment of how various key parameters influence the soil-structure interaction (SSI) response. To conduct SSI analysis, numerical methods that integrate boundary element (BE), finite element (FE), and scaled boundary finite element approaches are employed. A coupled FE-BE method was proposed by [7] for studying SSI issues in the time domain using real earthquake records. In this approach, a concrete dam-foundation system was divided into a near field and a far field, where the FE approach was used to model the near field and the BE approach was used for the far field. Equilibrium and compatibility conditions at the soil-structure interface linked these two methods. A hybrid approach, combining BE and FE, was used to analyze the dynamic interaction between soil and a cooling tower [8] with the numerical results compared to findings from full-scale tests. A dynamic SFSI study was presented by [9], where the soil was modeled with BE, and the structure and foundation were modeled with FE. The coupling of these elements at the soil-structure interface was achieved through equilibrium and compatibility conditions. The combined finite difference/finite element (FD/FE) method developed by [10] employed a FD in the time domain to compute the soil stresses and velocities for each cycle, and the displacement of soil grid points along the soil-structure boundary. These displacements were then used by the FE algorithm to calculate soil-structure nodal forces based on the stress state of the deformed structures, which were subsequently fed back into the FD code for use in the next time step. For frequency-domain solutions to dynamic SSI problems, ref. [11] investigated a coupled FE and fast BE approach. Numerical examples were used to assess the performance of direct, iterative, and monolithic coupling procedures. A coupled model combining FEM, BEM, and scaled boundary FEM was developed to assess the dynamic response of two-dimensional structures on layered soil media [12]. This model used a substructure method to link these approaches, and a computer program was created to perform the harmonic analysis of SSI systems. Using FEM and Minimax Probability Machine Regression (MPMR), ref. [13] studied the effects of the dynamic SSI on multi-story buildings with varying height-to-width ratios, modeling different soil types and conducting time history analysis for the El Centro and Bhuj earthquakes.