# Seismic Analysis of a Large LNG Tank Considering Different Site Conditions
## 2 Methodology  
### Smoothed Particle Hydrodynamics (SPH) Theory  
In the SPH method, the computational domain is dispersed into a series of randomly arranged particles. Adjacent particles interact based on smooth kernel functions in the forms of momentum exchange and energy transmission. The continuity equation can be expressed as:  
\[\frac{d\rho}{dt}+\rho\
abla\times\
u=0 \tag{1}\]  
The differential expression of the momentum equation is shown as:  
\[\rho\frac{dv}{dt}=\
abla P+\rho F \tag{2}\]  
The SPH algorithm is formulated as two parts: the integral representation of the field function and the particle approximation [20]. The integral representation of the function \(f(r)\) is identified as [21]:  
\[f(r)=\int_{\Omega}f(r^{\prime})\delta(r-r^{\prime})dr^{\prime} \tag{3}\]  
where \(f(r)\) represents a function of the three-dimensional position vector \(r\), and the Dirac delta function is written as:  
\[\delta(r-r^{\prime})= \begin{cases}1&(r=r^{\prime})\\ 0&(r\
eq r^{\prime})\end{cases}  \tag{4}\]If a Delta function kernel \(\delta(r-r^{\prime})\) is replaced by a smoothing function \(W(r-r^{\prime},h)\), the integral representation of \(f(r)\) can be shown as:  
\[f(r)\ \approx\ \int_{\Omega}f(r^{\prime})W(r-r^{\prime},h)dr^{\prime} \tag{5}\]  
where \(h\) is the smoothing length that defines the area of influence of the smoothing kernel function \(W\). \(W\) should satisfy three conditions:  
(1) the normalization condition stating:  
\[\int_{\Omega}W(r-r^{\prime},h)dr^{\prime}=1 \tag{6}\]  
(2) when the smoothing length approaches zero, the Delta function property is obtained as:  
\[\underset{h\to 0}{lim}W(r-r^{\prime},h)=\delta(r-r^{\prime}) \tag{7}\]  
(3) the compact condition implies that:  
\[W(r-r^{\prime},h)=0\ \text{when}\ \big{|}r-r^{\prime}\big{|}>\kappa h \tag{8}\]  
where \(\kappa\) represents a constant associated with the smoothing function of the point at \(r\) and defines the effective (non-zero) area for the smoothing function. In this study, a cubic spline is employed as the kernel function \(W\), and the expression of \(W\) can be shown as:  
\[W(r-r^{\prime},h)=a\left\{ \begin{array}{cc}\left(1-\frac{3}{2}q^{2}+\frac{3 }{4}q^{3}\right),&0\leq q\leq 1\\ \frac{1}{4}(2-q)^{3},&1\leq q\leq 2\\ 0,&2<q\end{array} \right. \tag{9}\]  
The movement of particles in SPH method generally follows the law of conservation of mass, the law of conservation of momentum, and the law of conservation of energy. A continuous density particle approximation method approximates the density using the SPH approximation concept to transform the continuity equation. By applying different conversions and operations, different forms of the density approximation equations can be determined. For instance, SPH approximation can be applied to the velocity divergence, and the particle density in the continuity equation can be obtained in the form of a gradient:  
\[\frac{D\rho_{i}}{Dt}=-\rho_{i}\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j}}y_{j}^{ \beta}\frac{\partial W_{ij}}{\partial r_{i}^{\beta}} \tag{10}\]  
Approximate expression of unit gradient SPH central particles is shown below:  
\[\
abla 1=\int 1\cdot\
abla W(r-r^{\prime},h)dr^{\prime}=\sum_{j=1}^{N}\frac{ m_{j}}{\rho_{j}}y_{j}^{\beta}\frac{\partial W_{ij}}{\partial r_{i}^{\beta}}=0 \tag{11}\]  
Therefore, another expression for the density approximation equation is obtained:  
\[\frac{D\rho_{i}}{Dt}=\rho_{i}\sum_{j=1}^{N}\frac{m_{j}}{\rho_{j}}y_{ij}^{\beta }\cdot\frac{\partial W_{ij}}{\partial r_{i}^{\beta}} \tag{12}\]The particle approximation method for the momentum equation closely resembles the continuous density method but requires some transformations. Based on different transformations, the momentum equation approximation can be derived in differential form. \[\sigma^{\alpha\beta}=-p\sigma^{\alpha\beta}+\tau^{\alpha\beta} \tag{13}\]  
where \(\sigma\) denotes total stress tensor, \(p\) represents isotropic pressure, and \(\tau\) is viscous pressure.