We can also see, from the literature review, that numerical methods such as the FEM and the BEM have been widely used (whether separately or combined) to study the complex soil-structure interaction problems. The former has a major disadvantage within satisfying the boundary condition at infinity, and artificial boundaries have been considered to solve this problem. However, the BEM does not need an artificial boundary because it uses Green’s functions that satisfy the radiation conditions implicitly at infinity. Indeed, attempting to solve numerically the boundary value problems for a multi-layered poroelastic medium numerically using BEM leads one to search for the associated Green’s function. Nevertheless, a discrete Green’s functions can be easily be provided for arbitrarily multi-layered solid medium stratum , as well as, a multi-layered poroelastic media stratum by a semi-analytical method like the Thin Layer Method (TLM).
The TLM allows an easy application of the BEM for the dynamic analysis of laminated media. In the intervening years, TLM has become an increasingly important tool for the engineering problems. It should be noted that the use of TLM for the treatment of unbounded medium was limited until the mid-1980s. The calculation of the Green’s function for the case of a layered half-space via TLM, was performed by the so-called paraxial approximation or Paraxial Boundary (PB). This laterThe latter is based on an algebraic approximation of the stiffness matrix of the elastic half-space. Once a second-order approximation is performed by calculating the first three terms of Taylor expansion in series, the algebraic approximation will completely absorb the waves propagating vertically or with lesser degrees with respect to the vertical axis. This combination between TLM-BEM and PB was well-developed and used by Guangyun et al. for the study of the active vibration isolation by wave impeding block in saturated media.