# A Dynamic Analysis Method of Liquid-Filled Containers Considering the Fluid-Structure Interaction
## 2 The FEA Theory Based on Acoustic Fluid Elements  
A liquid-filled container and a liquid constitute an FSI system, in which the liquid and structure (liquid-filled container) are simulated using FEM. The FEM simulation methods of structures have been introduced elsewhere [18]. For FEA considering the effect of FSIs, one should use the fluid displacement as an unknown quantity [19] and harness the similarity between the fluid motion equation and the equation of motion of structural elastomers, which results in a finite element calculation model of fluid consistent with the finite element scheme; the other is to take the fluid pressure as an unknown quantity [20] and coordinate the displacement and pressure on the structure-fluid interface, from which the obtained mass and stiffness matrices are asymmetric matrices. When analyzing liquid sloshingproblems using the pressure scheme based on acoustic fluid elements [21], the theory is as described below.  
For the structure,  
\[[M_{S}]\{\ddot{u}\}+[C_{S}]\{\dot{u}\}+[K_{S}]\{u\}-[R]\{p\}=\{f_{S}\} \tag{1}\]  
where \([M_{S}],[C_{S}],[K_{S}]\) separately represent the mass, damping, and stiffness matrices of the structure; \(\{\ddot{u}\},\{\dot{u}\},\{u\}\) separately denote the acceleration, speed, and displacement vectors at nodes of structural elements; \([R]\) denotes the coupling matrix at the structure-fluid interface; \(\{p\}\) is the nodal pressure vector of fluid elements; and \(\{f_{S}\}\) is the load vector of the structure.  
For the fluid,  
\[[M_{F}]\{\,\ddot{p}\}+[C_{F}]\{\dot{p}\}+[K_{F}]\{p\}+\vec{p}_{0}[R]^{T}\{\ddot {u}\}=\{f_{F}\} \tag{2}\]  
where \([M_{F}],[C_{F}],\) and\([K_{F}]\) separately represent the mass, damping, and stiffness matrices of the fluid; \(\{\dot{p}\},\{\ddot{p}\}\) are the first-order and second-order derivatives of nodal pressure of fluid elements; \(\vec{p}_{0}\) denotes the fluid density; and \(\{f_{F}\}\) is the load vector of the fluid.  
The following can be obtained by combining Equations (1) and (2):  
\[\begin{bmatrix}[M_{S}]&0\\ \vec{p}_{0}[R]^{T}&[M_{F}]\end{bmatrix}\begin{Bmatrix}\{\ddot{u}\}\\ \{\ddot{p}\}\end{Bmatrix}+\begin{bmatrix}[C_{S}]&0\\ 0&[C_{F}]\end{bmatrix}\begin{Bmatrix}\{\dot{u}\}\\ \{\dot{p}\}\end{Bmatrix}+\begin{bmatrix}[K_{S}]&-[R]\\ 0&[K_{F}]\end{bmatrix}\begin{Bmatrix}\{\{u\}\\ \{p\}\end{Bmatrix}=\begin{Bmatrix}\{f_{S}\}\\ \{f_{F}\}\end{Bmatrix} \tag{3}\]  
By solving Equation (3), the liquid sloshing modes in the FSI system and the FSI dynamic responses can be obtained. The bounding surface of the water domain are modeled using various types of interface defined as follows:  
1. _Fluid-structure interaction:_ Set a fluid-structure interaction on the boundary between fluid and structure.
2. _Free surface:_ Set a free surface on the top surface of water domain, which can provide an approximate representation of the water surface wave.
3. _Rigid wall:_ Set a rigid wall on the bottom and the lateral surface where the water cannot flow through the boundary.  
As explained above, the surface of the water domain is a defined, specified boundary condition. In general, the software ANSYS 2023 can automatically generate fluid-structure interactions along the boundary between the fluid and structure.