# Seismic Analysis of a Large LNG Tank Considering Different Site Conditions
## 3 Sloshing Frequency
### Details of the Small Tank Model  
A model of a small rectangular tank is developed to verify the accuracy the SPH-FEM algorithm and to determine the ratio of suitable element size to container size in a large LNG tank. The tank is 270 mm in length, 270 mm in width, and 300 mm in height, and contains water. The tank model is composed of glass with the following material properties: density \(p=950\) kg/m\({}^{3}\), Young's modulus \(E=1.173\times 10^{9}\) N/m\({}^{2}\), and Poisson ratio \(v=0.35\). Liquid sloshing is an extremely complex behaviour, and an equation of state is applied to define the pressure, volume, and energy of the material. In this section, the Newtonian fluid constitutive material formulation is used for water. The Mie-Gruneisen equation of state defined in Table 1 describes the volumetric strength and density of a material. The isotropic pressure in the Mie-Gruneisen equation of state is stated as follows:  
\[\left\{\begin{array}{l}p(\rho,e)=\left(1-\frac{1}{2}\Gamma_{0}\eta\right)P_ {H}+\Gamma_{0}\rho_{0}E_{m}\\ p_{H}=\frac{\rho_{0}c^{2}\eta}{\left(1-\eta n\right)^{2}}\\ \eta=\frac{\rho}{\rho_{0}}-1\end{array}\right. \tag{17}\]  
where \(\eta\) is the volumetric compressibility, \(H\) is the Hugoniot curve, \(E_{m}\) is the internal energy per unit mass, and \(c\) is the sonic speed of the material. The linear relationship between the shock velocity and the particle velocity can be used to describe the coefficients \(s\) and \(c\):  
\[U_{s}=c+sU_{p} \tag{18}\]  
where \(U_{P}\) is the particle velocity and \(U_{s}\) is the shock wave velocity. This equation of state describes the fluid motion.  
A model for the hydrodynamic material is provided by the equation of state \(U_{s}-U_{p}\), and the properties of water are shown in Table 1. In the model, only water is discretized as SPH particles. The SPH domain has a length of 270 mm on one side, a width of 270 mm on the other side, and a height ranging from 50 to 200 mm. The exterior Lagrangian mesh is composed of 5042 hexahedral elements in ABAQUS [24; 25]. The entire simulation lasts for 2.76 s, and a horizontal base displacement is introduced with a constant velocity of 0.75 m/s in the first 0.2 s as shown in Figure 2.  
The SPH algorithm employs a cubic spline as the interpolation polynomial. Smoothing length is an important parameter and has significant influence on the accuracy of the prediction [26; 27]. By default, ABAQUS calculates a smoothing length at the beginning of an analysis where an element is associated with an average of approximately 30 to 50 particles. Every increment recomputes this local connectivity internally and computes kinematic quantities (normal and shear strains, deformation gradients, etc.) based on contributions from a cloud of particles centered around the particle of interest. Subsequently, stress is computed in a similar fashion as reduced-integration brick elements, which are in turn used to compute element nodal forces for the particles in the cloud based on the SPH formulation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{**Density (kg/m\({}^{3}\))**} & \multirow{2}{*}{**Sonic Speed (m/s)**} & \multirow{2}{*}{**Dynamic Viscosity (kg/(m/s)**} & \multirow{2}{*}{**Coefficients of the Equation of State**} \\ \cline{5-5}  & & & & \(s\) & \(\Gamma_{0}\) \\ \hline Water & 1000 & 1480 & 0.001 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of water.  
In this study, a constant smoothing length is defined for the entire analysis. This modification is compliant with ABAQUS [22] standards and improves the quality of simulation.