## 2 Modelling of Fluid-Tank-Foundation System Based on Lagrangian Approach  
Fluid-structure interaction problems can be investigated by different methods, such as the added mass method, Lagrangian method, Eulerian method, and Lagrangian-Eulerian method. In this study, the displacement-based Lagrangian method is selected to simulate the interaction of the tank and fluid. The fluid element is defined by eight nodes, and each node has three degrees of freedom: translation in the X, Y, and Z directions of the node. Because of the lack of geometric capability in the Lagrangian finite element method considering the brick elements, the intze-type is idealized as a cylindrical vessel that has the same capacity with the intze-type [15]. When the fluid in the tank structure vibrates, the free surface always tends to the equilibrium position under the action of gravity. To simulate the effect of the restoring force on the free surface of the fluid, an additional spring must be considered in the free demonstration. A special surface effect should be considered when selecting the fluid element option; that is, adding a gravity spring to each node to restore the surface to the equilibrium position, as shown in Figure 1. The parameters of the gravity spring are positive at the top of the element and negative at the bottom of the element. If there is a free surface, the gravity effect must be included. For internal nodes, the positive and negative effects cancel each other. At the bottom, the fluid must be contained to prevent fluid leakage, and the negative spring does not work (as long as all degrees of freedom at the bottom are fixed) [16]. The positive stiffness of the spring can be described as  
\[\mathbf{K_{s}}=\rho A_{\mathrm{f}}(g_{x}\mathbf{C_{x}}+g_{y}\mathbf{C_{y}}+g_{ z}\mathbf{C_{z}}), \tag{1}\]  
where \(\rho\) is the mass density, \(A_{\mathrm{f}}\) is the surface area of the component, and \(g_{i}\) and \(\mathbf{C_{i}}\) are the acceleration in the direction of the \(i\) and \(i\)th component of the normal to the face of the element, respectively. The mass and stiffness matrix of the fluid element can be expressed as  
\[\mathbf{M_{f}}=\rho\int_{\mathrm{v}}\mathbf{Q}^{T}\mathbf{Q}dV=\rho\sum_{i} \sum_{j}\sum_{k}\eta_{i}\eta_{j}\eta_{k}\mathbf{Q}_{ijk}^{T}\mathbf{Q}_{ijk} \mathrm{det}\mathbf{f}_{ijk}, \tag{2}\]  
\[\mathbf{K_{f}}=\int_{\mathrm{v}}\mathbf{B}^{T}\mathbf{E}\mathbf{B}dV=\sum_{i} \sum_{j}\sum_{k}\eta_{i}\eta_{j}\eta_{k}\mathbf{B}_{ijk}^{T}\mathbf{E}\mathbf{ B}_{ijk}\mathrm{det}\mathbf{f}_{ijk}, \tag{3}\]  
Figure 1: Finite element model of the fluid–tank–foundation system and a schematic diagram of the fluid–structure coupling. where \(\mathbf{Q}_{ijk}\) is the interpolating function, \(\eta_{i}\), \(\eta_{j}\) and \(\eta_{k}\) are the weight functions, \(\mathbf{J}\) is the Jacobian matrix, \(\mathbf{B}\) is the strain-displacement matrix, and can be computed from \(\varepsilon=\mathbf{B}\mathbf{u}\). The equations of kinetic energy and potential energy can be given as  
\[U=\Pi_{\varepsilon}=\frac{1}{2}\mathbf{u}^{\mathrm{T}}\mathbf{K}_{\mathrm{f}} \mathbf{u}, \tag{4}\]  
\[T=\frac{1}{2}\mathbf{v}^{\mathrm{T}}\mathbf{M}_{\mathrm{f}}\mathbf{v}, \tag{5}\]  
Substituting the above expressions of kinetic energy and potential energy into the Lagrange equation gives  
\[\frac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\frac{\partial T}{\partial\dot{u}_{j}} \bigg{)}-\frac{\partial T}{\partial u_{j}}+\frac{\partial U}{\partial u_{j}}= F_{j}, \tag{6}\]  
where \(u_{\mathrm{j}}\) is the \(j\)th displacement component and \(F_{\mathrm{j}}\) is the external load applied to the system. The governing equation of the system can be expressed as  
\[\mathbf{M}_{\mathrm{f}}\ddot{\mathbf{u}}+(\mathbf{K}_{\mathrm{f}}+\mathbf{K }_{\mathrm{s}})\mathbf{u}=\mathbf{R}, \tag{7}\]  
where \(\ddot{\mathbf{u}}\) is the acceleration and \(\mathbf{R}\) is a general time-varying load vector.