## 2 Subdomain Partition Method for Fluid  
A cylindrical tank resting on soil is partially full of incompressible, irrotational and inviscid liquid in Figure 1. Multiple rigid annular baffles are positioned on the tank wall. The soil foundation is considered as the half space. The rigid tank is supported on the circular base whose radius is the same as that of the storage tank. The cylindrical coordinate system with the origin located at the bottom center is employed to describe the movement for the soil-tank system undergoing a horizontal motion. Due to small amplitude for free surface under small earthquake, the linear liquid sloshing is considered. The influences of the baffle thickness on the sloshing can be overlooked. The number of anti-sloshing baffles is considered as \(M\). \(R_{1}\) and \(R_{2}\) are radiuses of the baffles and tank, respectively. The baffles are, respectively, positioned at \(z=h_{1},\ h_{2},\ \ldots,\ h_{M}.\ h_{M+1}\) and \(h_{0}\) are the fluid height and the tank bottom, respectively. According to the subdomain partition method [17; 18], the subdomains of liquid \(\Omega_{i}\ (i=1,\ 2,\ \ldots,\ 2M+2)\) are determined with \(M\) circular and \(M+1\) cylindrical interfaces, as shown in Figure 2. The velocity potential for fluid yields  
\[\phi(r,\ \theta,\ z,\ t)=\phi_{i}(r,\ \theta,\ z,\ t),\ (r,\ \theta,\ z)\in \Omega_{i},\ (i=1,\ 2,\ \ldots,\ 2M+2) \tag{1}\]  
where \(\Gamma_{2M+2}\) and \(\Gamma_{2M+3}\) are free surfaces for subdomains \(\Omega_{2M+2}\) and \(\Omega_{2M+1}\), respectively. The \(M\) circular artificial interfaces from bottom baffle to top one are identified as \(\Gamma_{M+2},\ \Gamma_{M+3},\ \ldots,\ \Gamma_{2M+1}.\) The \(M+1\) cylindrical artificial interfaces from bottom to top are identified as \(\Gamma_{1},\ \Gamma_{2},\ \ldots,\ \Gamma_{M+1}.\)  
The liquid velocity potential \(\phi_{i}\) satisfies Laplace equation based on assumptions and definitions:  
\[\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi_{i}}{\partial r})+ \frac{1}{r^{2}}\frac{\partial^{2}\phi_{i}}{\partial\theta^{2}}+\frac{\partial ^{2}\phi_{i}}{\partial z^{2}}=0,\ (i=1,\ 2,\ \ldots,\ 2M+2) \tag{2}\]  
\(\Gamma_{l}\) denotes the artificial interface between the adjacent subdomains \(\Omega_{i}\) and \(\Omega_{l^{\prime}}\ (i<i^{\prime})\) which meet continuity conditions of pressure and velocity at surface \(\Gamma_{l}\), respectively:  
\[\frac{\partial\phi_{i}}{\partial t}=\frac{\partial\phi_{i^{\prime}}}{\partial t },\ \frac{\partial\phi_{i}}{\partial\mathbf{n}_{l}}\ =\frac{\partial\phi_{l^{\prime}}}{\partial\mathbf{n}_{l}},\ (l=1,\ 2,\ \ldots,\ 2M+1) \tag{3}\]  
Figure 1: Soil–tank coupling system. Figure 2: Artificial surfaces and liquid subdomains. where \(\mathbf{n}_{l}\) is the normal vector for \(\Gamma_{l}\). The normal vector for the velocity potential function should be consistent with that for the boundary due to no liquid cavitation at interfaces. Considering free surface waves, the movement equation yields  
\[\left.\frac{\partial\phi_{l}}{\partial t}\right|_{z=h_{M+1}}+gf_{i}=0,\ (i=2M+1,\ 2M+2) \tag{4}\]  
where \(g\) is the gravity acceleration. \(f_{i}\) represents the sloshing height for \(\Omega_{i}\) and has  
\[f_{i}=\int_{0}^{t}\frac{\partial\phi_{l}}{\partial z}\bigg{|}_{z=h_{M+1}} \mathrm{d}t,\ (i=2M+1,\ 2M+2) \tag{5}\]  
Regard \(\ddot{u}(t)\) as horizontal absolute acceleration of the cylindrical container. The liquid can be regarded as the impulsive component moving with the rigid wall and convective one undergoing the sloshing [6]. The velocity potential is the sum of the impulsive velocity potential \(\phi_{l}^{\mathrm{I}}(r,\theta,z,t)\) and convective velocity potential \(\phi_{l}^{\mathrm{C}}(r,\theta,z,t)\), namely, \(\phi_{i}(r,\theta,z,t)=\phi_{i}^{\mathrm{I}}(r,\theta,z,t)+\phi_{i}^{\mathrm{C }}(r,\theta,z,t)\). Combined with Equations (2)-(5), \(\phi_{i}^{\mathrm{I}}(r,\theta,z,t)\) and \(\phi_{i}^{\mathrm{C}}(r,\theta,z,t)\) should meet the movement equation, continuity and boundary conditions, respectively.