## 1 Introduction
However, unlike the classical approach, though the Lagrange multipliers are introduced as the interface forces, the IBS method does not introduce additional variables. After the derivation of the coupling functions, Lagrange multipliers become the only direct unknown variables, and the solution of IBS becomes concise and easy to realize (detailed in Section 2). Another class of methods for the non-matching interfaces problem refers to the construction of transition elements. [PERSON] [20] proposed a joint/interface element to join typical problems involving shell-to-shell and shell-to-solid interfaces. The construction of the proposed element follows the usual finite element procedure, and the stiffness matrix is obtained by the standard procedure of minimizing the total potential energy. [PERSON] and [PERSON] [21] developed a zero-thickness element considering the elastoplastic constitutive law. [PERSON] [22] constructed an interface element based on the Mindlin-Reissner plate assumptions to couple solid and shell elements. However, for these approaches based on variational principles and constructing transition elements, one needs to access the shape functions of the elements over the interface, which is intrusive and complicated. The third class of methods considers an interpolation scheme inspired by meshless methods. [PERSON] [23] proposed an interface element method (IEM), the shape functions of which are constructed by the implementation of the moving least squares (MLS) approximations. [PERSON] and [PERSON] [24; 25] proposed a displacement welding technique using a blended function based on the MLS function, benefiting from the meshless character of the MLS function, which can be utilized in coupled analysis of FE substructures with mismatching interfaces. [PERSON] and [PERSON] [26] compared different methods (nearest-neighbor interpolation, weighted residual methods and radial basis functions) that can deal with the information transfer between non-matching meshes in fluid-structure interaction computations for different criteria. Another emphasis of the interpolation scheme for non-matching interfaces is the interpolation method. The normally used interpolation methods are splines, Kriging [27] and radial basis functions [28; 29]. Benefiting from consistency and the Kronecker delta function property, the Kriging interpolation function was widely used both in structural dynamics and fluid dynamics [30; 31]. The assembly of FE models with non-matching interfaces can be achieved by the above-mentioned approaches. However, the framework of the IBS method is quite different from the finite element method (FEM). The IBS method is a reduced time-domain substructure method: IRFs can be regarded as \"super elements in time\" [12], and only limited input-output points are retained [13]. The assembly of substructure IRFs is based on the displacement compatibility of the limited discrete interface nodes, which are described directly in the physical domain [5; 8]. Therefore, it is essential to study the scheme for the assembly of substructure IRFs with non-matching interfaces. To couple non-matching interface substructure IRFs for transient dynamic prediction, we use the normalized interpolation scheme and form an extended framework of the IBS method. Numerical models with efficient interpolation functions with normalization conditions (such as the moving Kriging interpolation function or the MLS function) are used to evaluate the proposed framework. The aim of this paper is to propose a multi-domain substructure synthesis method. We consider the ordinary IBS method and present a method for coupling non-matching interfaces between substructures, including IBS, FEM and modal substructures. The work of this paper is organized as follows: First, the basic concept of the IBS method is briefly reviewed. In Section 2, the process of coupling a multi-domain substructure with non-matching interfaces is presented. In Section 4, the basic interpolation rule is detailed.