## 2 SPH Methods  
The movement of inner water in an arch rib under seismic vibration is a sloshing problem. The viscosity of water can be ignored in this study. The governing equations of fluid dynamics can be reduced from the Navier-Stokes (N-S) equations to Euler equations. SPH is a purely Lagrangian method, so the following Lagrangian form continuity and momentum equations are often used in SPH representations:  
\[\frac{\mathrm{D}\rho}{\mathrm{D}t}=-\rho\
abla\cdot\mathbf{u} \tag{1}\]  
\[\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}=-\frac{1}{\rho}\
abla\mathrm{p}+ \mathbf{g} \tag{2}\]  
where \(\rho\) is the fluid particle density; \(t\) is the time; \(\mathbf{u}\) is the particle velocity vector; \(\mathrm{p}\) is the pressure; and \(\mathbf{g}\) is the gravitational acceleration; \(\
abla=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{ \partial y}\) is the Nabla operator, and \((\mathbf{i},\mathbf{j})\) is the orthogonal basis of 2D Cartesian coordinates. There are various forms of the SPH formulation for the above governing equations, and they demonstrate both advantages and disadvantages depending on the different hydrodynamic applications. In this paper, we adopted a revised SPH representation proposed by [PERSON] and [PERSON] [19]. The SPH continuity equation and momentum equation are:  
\[\frac{\mathrm{D}\rho_{\mathrm{a}}}{\mathrm{D}t}=\rho_{\mathrm{a}}\sum_{ \mathrm{b=1}}^{\mathrm{N}}\frac{\mathrm{m}_{\mathrm{b}}}{\mathrm{p}_{\mathrm{b }}}\mathbf{u}_{\mathrm{ab}}\cdot\
abla_{\mathrm{a}}\mathrm{W}_{\mathrm{ab}} \tag{3}\]  
\[\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t}=-\frac{1}{\rho}\
abla\mathrm{p}+ \mathbf{g} \tag{4}\]  
where a is the number of reference particles and b represents the number of neighboring particles; N is the total number of neighboring particles of the reference particle a; m is the particle mass; \(\mathbf{u}\) is the velocity vector; \(\mathbf{u}_{\mathrm{ab}}\triangleq\mathbf{u}_{\mathrm{a}}-\mathbf{u}_{ \mathrm{b}}\) is the velocity difference between the reference particle a and its neighboring particle b and \(\
abla_{\mathrm{a}}\mathrm{W}_{\mathrm{ab}}\triangleq\
abla_{\mathrm{r}} \mathrm{W}(\left|\mathbf{r}-\mathbf{r}_{\mathrm{b}}\right|,\mathrm{h})\right|_ {\mathbf{r}=\mathbf{r}_{\mathrm{a}}}\) represents the gradient of a kernel taken with respect to a particle a. Here, \(\mathrm{W}\) is the kernel function for SPH interpolation. In this work, a modified Gaussian kernel proposed by [PERSON] and [PERSON] [19] is adopted:  
\[\mathrm{W}(\mathrm{s},\mathrm{h},\delta)=\frac{\mathrm{e}^{-(\mathrm{s}/ \mathrm{h})^{2}}-\mathrm{e}^{-(\mathrm{s}/\mathrm{h})^{2}}}{2\pi\int_{0}^{ \mathrm{s}}\mathrm{s}\left[\mathrm{e}^{-(\mathrm{s}/\mathrm{h})^{2}}-\mathrm{ e}^{-(\mathrm{s}/\mathrm{h})^{2}}\right]\mathrm{ds}} \tag{5}\]  
where \(\mathrm{s}\triangleq|\mathbf{r}-\mathbf{r}^{\prime}|\); \(\mathbf{r}\triangleq(x,y)\) represents the coordinate vector of the reference particle; \(\mathbf{r}^{\prime}\) is the coordinate vector of the neighboring particle; \(\mathrm{h}\) is the smoothed length, which is a measure of the support domain of the kernel function and is taken as a multiple of the initial particle spacing; and \(\delta\) represents the cut-off limit due to the original Gaussian function is not compact support. Here, we choose \(\delta=3\mathrm{h}\) as the compact support distance. According to our computational experience, this kernel has the advantage of good accuracy, better stability, and reasonable computational cost. In this work, the standard weakly compressible SPH method (WCSPH) is adopted. In the WCSPH, the fluid pressure is computed using the following equation of state:  
\[\mathrm{p}=\mathrm{B}\left[\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}-1\right] \tag{6}\]where \(\gamma=7\) and \(\rho_{0}=1000\) kg/m\({}^{3}\) are used for the water. The constant B is related to the sound speed c, which should be ten times or larger than the highest bulk flow velocity. It can be estimated by \(\mathrm{B}=\rho c^{2}/\gamma\).