# Numerical Algorithm for Dynamic Impedance of Bridge Pile-Group Foundation and Its Validation
## 1 Introduction
[24] presented a BEM-FEM model for the time-harmonic dynamic analysis of piles and pile groups embedded in an elastic half-space, and successfully reduced the number of degrees of freedom. [PERSON] and [PERSON] [25] developed a general three-dimensional finite element procedure for the dynamic impedances of pile groups with caps embedded in isotropic homogeneous elastic soils. [PERSON] et al. [26] investigated the dynamic behavior of a laterally loaded fixed-head pile group in a transversely isotropic multilayered half-space by BEM-FEM coupling method. With the porous medium model and the Rayleigh-Love rod model describing the soil and the pile groups, respectively, [PERSON] et al. [27] developed a simplified analytical solution for the vertical impedance function of viscoelastic pile groups partially embedded in a layered, transversely isotropic, liquid-saturated, viscoelastic soil medium. [PERSON] et al. [28; 29] proposed a numerical model for the 3D kinematic interaction analysis of vertical and inclined pile groups in horizontally layered soils, based on BEM-FEM coupling, in which piles are modelled with finite beam elements while the soil is schematized with infinite independent horizontal layers. [PERSON] et al. [30] presented a hybrid analytical-numerical method for the dynamic analysis of piles and pile groups embedded in nonhomogeneous transversely isotropic media subjected to lateral excitations, in which the piles are simulated using finite elements and the interactions between piles and soil are captured through a system of rigid, massless elements. [PERSON] et al. [31] proposed a substructure-method-based soil-structure interaction analysis model to study the vertical and horizontal dynamic impedances of piled rafts in homogeneous and layered soil conditions. [PERSON] and [PERSON] [32] developed a method to compute the dynamic response of pile groups in layered soils with compressible constituents by combining the analytical layer element method and finite element method. [PERSON] et al. [33] presented an analytical method to study the horizontal dynamic response of pile groups embedded in partially saturated soil based on the motion theory of three-phase poroelastic media, with the pile-soil system modeled using the Timoshenko beam theory and soil resistance incorporated using [PERSON]'s plane strain model. Almost all of the abovementioned methods possess the trait of strong specialty, the theoretical derivation procedures of which are complex and even involve programming processes. In view of the complexity of actual soil layers and the parametric diversity of bridge group foundations, it can be found that the above calculation methods are too complex and specialized to be generalized in practical projects. Therefore, this paper presents a project-oriented numerical algorithm for the dynamic impedance of bridge pile-group foundation. Based on the viscous-spring artificial boundary theory, the derivation and solution of the impedance function are transferred to numerical modeling and harmonic analysis, which can be carried out using the finite element method. The applicability of the proposed numerical algorithm is verified by comparing its calculation results with those obtained through methods in existing literature and a field experiment. Both comparisons show that the numerical algorithm satisfies the project demand of analysis precision, thus showing good effectiveness in application.