# Dynamic Responses of Liquid Storage Tanks Caused by Wind and Earthquake in Special Environment
## 4 Fluid-Solid Interaction
### Shell-Liquid Interaction  
Because the calculation process involves a large number of nonlinearities, the exact solution of each response can be obtained through multiple equilibrium iterations. \(\Delta\phi\) is used to express the increment of the unknown velocity potential \(\mathbf{\phi}\), and \(\Delta\mathbf{u}\) is used to express the increment of the unknown displacement \(\mathbf{u}\). The shell-liquid interaction dynamic equation based on potential fluid theory is [24]  
\[ \begin{array}{l}\left[\begin{array}{cc}\mathbf{M}_{\text{ss}}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}_{ll}\end{array}\right]\left[\begin{array}{c}\Delta \ddot{\mathbf{u}}\\ \Delta\dot{\mathbf{\phi}}\end{array}\right]+\left[\begin{array}{cc}\mathbf{C }_{\mathbf{uu}}+\mathbf{C}_{\text{ss}}&\mathbf{C}_{\mathbf{u}l}\\ \mathbf{C}_{\mathbf{lu}}&-(\mathbf{C}_{ll}+(\mathbf{C}_{ll})_{S})\end{array} \right]\left[\begin{array}{c}\Delta\dot{\mathbf{u}}\\ \Delta\dot{\mathbf{\phi}}\end{array}\right]+\\ \left[\begin{array}{cc}\mathbf{K}_{\mathbf{uu}}+\mathbf{K}_{\text{ss}}& \mathbf{K}_{\mathbf{lu}}\\ \mathbf{K}_{\mathbf{ul}}&-(\mathbf{K}_{ll}+(\mathbf{K}_{ll})_{S})\end{array} \right]\left[\begin{array}{c}\Delta\mathbf{u}\\ \Delta\dot{\mathbf{\phi}}\end{array}\right]=\left[\begin{array}{cc}\mathbf{ F}_{\text{ss}}\\ \mathbf{0}\end{array}\right]-\left[\begin{array}{c}\mathbf{F}_{p}\\ \mathbf{F}_{l}+(\mathbf{F}_{l})_{S}\end{array}\right]\end{array}  \tag{17}\]  
where \(\mathbf{M}_{ll}\) is the liquid mass matrix; \(\mathbf{C}_{uu},\mathbf{C}_{lu}\), \(\mathbf{C}_{ul}\), and \(\mathbf{C}_{ll}\) are the damping matrices of the structure itself, the liquid contributed by the structure, the structure contributed by the liquid and the liquid itself, respectively; and \(\mathbf{K}_{uu}\), \(\mathbf{K}_{\mathbf{lu}}\), \(\mathbf{K}_{\mathbf{ul}}\), and \(\mathbf{K}_{ll}\) are the stiffness matrices of the structure itself, the liquid contributed by the structure, the structure contributed by the liquid and the liquid itself, respectively; \(\mathbf{F}_{p}\), \(\mathbf{F}_{l}\), and \((\mathbf{F}_{l})_{S}\) are the forces acting on the structural boundary caused by the liquid pressure, volume force, and area force, respectively; \(\mathbf{F}_{l}\) is obtained by the volume integral of Equation (18), and \((\mathbf{F}_{l})_{S}\) is obtained by surface integral of Equation (19) [24]  
\[\mathbf{F}_{l}=\int\limits_{V}\left(\frac{\partial\rho_{l}}{\partial\hbar}\dot {h}\delta\phi-\rho_{l}V\phi\right)dV \tag{18}\]  
\[\left(\mathbf{F}_{l}\right)_{S}=\int\limits_{S}-\rho_{l}\mathbf{u}\cdot \mathbf{n}\delta\phi dS \tag{19}\]  
where \(\rho_{l}\) is the liquid density; \(V\) is the liquid domain; \(S\) is the liquid domain boundary; \(\mathbf{n}\) is the internal normal direction vector of \(S\); and \(\dot{\mathbf{u}}\)Velocity and acceleration of Equation (22) at \(t+\Delta t\) can be expressed as functions of unknown displacement  
\[ \begin{split} t+\Delta t\
u&=\frac{1}{\alpha\Delta t} \Big{(}t+\Delta t\,u-\,^{t}u\Big{)}-\,^{t}\
u\Big{(}\frac{1}{\alpha}-1\Big{)}= \,^{t+\Delta t}dm+\,^{t}\xi\\ t+\Delta t\,a&=\frac{1}{\alpha^{2}\Delta t^{2}} \Big{(}t+\Delta t\,u-\,^{t}u\Big{)}-\,^{t}\
u\frac{1}{\alpha^{2}\Delta t}-\,^{t }a\Big{(}\frac{1}{\alpha}-1\Big{)}=\,^{t+\Delta t}dn+\,^{t}\eta\end{split}  \tag{23}\]  
Taking Equations (22) and (23) into wind field equation \(G_{w}\Big{[}w,\dot{w}\Big{]}=0\) and structure equation \(G_{s}\Big{[}u,\dot{u},\ddot{u}\Big{]}=0\)  
\[ \begin{split} t+\alpha\Delta t\,G_{w}\approx G_{w}\Big{[}t+ \alpha\Delta t\,w,\Big{(}t+\alpha\Delta t\,w-\,^{t}w\Big{)}/\alpha\Delta t\, \Big{]}=0\\ t+\Delta t\,G_{s}\approx G_{s}\Big{[}t+\Delta t\,u,\,^{t+ \Delta t}dm+\,^{t}\xi,\,^{t+\Delta t}dn+\,^{t}\eta\Big{]}=0\end{split}  \tag{24}\]  
In order to solve the coupled system, Equation (24) is discretized. Assuming that the solution vector of the coupled system is \(\mathbf{X}=\mathbf{X}(\mathbf{X}_{w},\,\mathbf{X}_{s})\), \(\mathbf{X}_{w}\), and \(\mathbf{X}_{s}\) represents solution vectors of wind field and structure nodes.