# A Dynamic Analysis Method of Liquid-Filled Containers Considering the Fluid-Structure Interaction
## 1 Introduction  
The effect of fluid-structure interactions (FSIs) between solid structures and liquids is prevalent in engineering practice. For example, liquid-filled containers under seismic action or other vibration loads are a structure with the typical effect of FSIs [1]. Analysis of the FSI effect of liquid-filled containers, on the one hand, focuses on liquid sloshing therein, and on the other, considers the influence of liquid sloshing on these containers. In order to tackle liquid sloshing problems, [PERSON] et al. [2; 3; 4; 5] systematically expounded the theoretical and engineering applications of liquid sloshing modes. However, these methods pay more attention to theoretical and analytical methods and are only applicable to solving cases with regular shapes and simple external excitations, yet fail to solve vibration problems of liquid-filled structures of complex shapes [6; 7]. [PERSON] et al. [8; 9] studied liquid modes based on potential-based fluid elements. When using this method, the liquid sloshing frequency is much lower than the structural vibration frequency, and the resulting first hundreds of modes are all liquid sloshing modes rather than modes of solid structures [10]. Therefore, it is difficult to apply the mode-superposition response spectrum method and time history analysis to any dynamic analysis of liquid-filled structures, and these methods are inapplicable to the analysis of dynamic responses of liquid-filled containers. Structural engineers generally pay more attention to how liquid motion affects structural responses under seismic action, while they are not interested in the motion of liquids themselves. To consider the influence of liquid sloshing on the dynamic responses of liquid-filled containers and avoid complex liquid sloshing computation, the simplified equivalent mechanical model of fluid sloshing is generally used. On the basis of the potential flow theory, [PERSON] [11] took the lead to establish an equivalent mechanical model of fluid sloshing in a rectangular container, which has been widely applied in the engineering field. [PERSON] [12] deduced a simplified calculation formula based on an equivalent model of fluids based on physical intuition. The Housner model is a good approximation for the exact solution to the model proposed by [PERSON] and has found extensive applications in civil and hydraulic engineering; however, the [PERSON] model is not based on physical intuition and therefore is not an exact physical model, so its results are unreliable in some cases. [PERSON] [13] improved the Housner model on the basis of the theory of linear potential flow and provided a fitting solution to the equivalent model using a semi-analytical and semi-numerical method, whereas the improved Housner model has such complex parameters that it is only suitable for regular two-dimensional (2D) models and not complex 2D and three-dimensional (3D) models. Apart from proposing the equivalent methods based on the potential flow theory, [PERSON] [14] and [PERSON] [15] also developed the added-mass method. [PERSON] et al. [16] applied the added-mass method to the finite element method (FEM) and performed FSI analysis. [PERSON] et al. [17] proposed an improved added-mass model based on the added-mass method and conducted dynamic analysis on an annular tank, providing reference for the design and application thereof. However, the distributed mass coefficient is difficult to determine in the above added-mass method and improved added-mass methods, so when using these methods to analyze structures of liquid-filled containers, the results are generally less reliable. The conventional analytical methods are only applicable to cases with regular geometric shapes and simple external excitations, and FEM based on potential-based fluid elements is not applicable to the analysis of the dynamic responses of liquid-filled structures. Moreover, equivalent mechanical models based on fluid sloshing and the added-mass methods also have drawbacks.