# A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures
## 1 Introduction  
The modeling of a complex structure and the conduction of accurate dynamics analysis is always time-consuming. The established model based on a traditional finite element method is not efficient in calculations, especially for large and complex structures, such as spacecraft and aircraft structures. In addition, various uncertain factors in the modeling process leads to errors in the computational results. Therefore, it is necessary to propose methods to improve modeling efficiency while considering modeling accuracy. For complex and huge-degrees-of-freedom systems, a substructure method is a good choice [1, 2]. The substructure is obtained by projecting the dynamics of a linear finite element model to a low-dimensional subspace spanned by so-called component modes. The component mode can be a collection of various modes, such as a rigid body mode, a normal mode, and a constrained mode [3, 4]. In recent years, the introduction and application of substructure methods in the static and dynamic analysis of different structural systems have been developed [5, 6]. [PERSON] et al. [7, 8, 9] used the improved substructure component mode synthesis based on the extremum response surface method to improve the computationalefficiency and reliability of atomized turbine blade disc vibration characteristics. With the help of substructure technology, [PERSON] [10]used the wave packet method to effectively evaluate the heat conduction of each phonon mode in the large-period harmonic lattice. [PERSON] et al. [11] developed a model simplification technique which can efficiently calculate and accurately analyze the forced response displacement and stress amplitude levels of a blade disc that vibrates in the high frequency range. [PERSON] et al. [12] used a hybrid-interface method based on component mode synthesis to develop a reduced-order model (ROM) for tuning systems with splines. Using these valid ROMs in forced response analysis, it was found that the new modeling method could save computational costs, while ensuring good accuracy compared to full-order finite element analysis. [PERSON] et al. [13] proposed an improved substructure-based response sensitivity method to accelerate the convergence rate of model updating. The accuracy and efficiency of the proposed method was tested by two numerical simulations and an experimental case. [PERSON] et al. [14; 15; 16] used a new iterative reduced-order substructure method to calculate the eigen-solutions and eigen-sensitivities of the global structure. By using this method, the eigen-solutions and eigen-sensitivities of the whole structure can be accurately calculated with only a few substructural eigenmodes and several iterations. [PERSON] et al. [17; 18] proposed a substructure-based finite element model updating technique. This method was applied to frame structures and actual Bridges, and the computation of traditional model updating methods performed on global structures was reduced. [PERSON] et al. [19] proposed a new shell super-element for studying linear/nonlinear static and free vibration analysis of spherical structures with partial or complete spherical geometry. The results showed that the supercell can achieve high precision with low computational costs. [PERSON] et al. [20] used a new cylindrical super-element for the design and vibration analysis of FGM rings to obtain the natural frequencies and modal shapes, and compared the results with simple cases obtained from analytical solutions and conventional elements. Considering a few newly designed super-elements, comparable results for simple cases in the reported literature can be achieved. [PERSON] et al. [21] proposed several models to analyze the load distribution and the deformation of slewing support components in wind turbine generators. This work presented a super-element-based FE model that can significantly reduce computational costs with a negligible loss of precision. [PERSON] et al. [22; 23] proposed a new substructure order reduction method with high computational efficiency.